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+1
+Learning Deep MRI Reconstruction Models from
+Scratch in Low-Data Regimes
+Salman Ul Hassan Dar, S¸ aban ¨Ozt¨urk, Muzaffer ¨Ozbey, and Tolga C¸ ukur∗
+Abstract— Magnetic resonance imaging (MRI) is an es-
+sential diagnostic tool that suffers from prolonged scan
+times. Reconstruction methods can alleviate this limitation
+by recovering clinically usable images from accelerated ac-
+quisitions. In particular, learning-based methods promise
+performance leaps by employing deep neural networks as
+data-driven priors. A powerful approach uses scan-specific
+(SS) priors that leverage information regarding the underly-
+ing physical signal model for reconstruction. SS priors are
+learned on each individual test scan without the need for
+a training dataset, albeit they suffer from computationally
+burdening inference with nonlinear networks. An alterna-
+tive approach uses scan-general (SG) priors that instead
+leverage information regarding the latent features of MRI
+images for reconstruction. SG priors are frozen at test
+time for efficiency, albeit they require learning from a large
+training dataset. Here, we introduce a novel parallel-stream
+fusion model (PSFNet) that synergistically fuses SS and
+SG priors for performant MRI reconstruction in low-data
+regimes, while maintaining competitive inference times to
+SG methods. PSFNet implements its SG prior based on a
+nonlinear network, yet it forms its SS prior based on a linear
+network to maintain efficiency. A pervasive framework for
+combining multiple priors in MRI reconstruction is algo-
+rithmic unrolling that uses serially alternated projections,
+causing error propagation under low-data regimes. To alle-
+viate error propagation, PSFNet combines its SS and SG
+priors via a novel parallel-stream architecture with learn-
+able fusion parameters. Demonstrations are performed on
+multi-coil brain MRI for varying amounts of training data.
+PSFNet outperforms SG methods in low-data regimes, and
+surpasses SS methods with few tens of training samples.
+In both supervised and unsupervised setups, PSFNet re-
+quires an order of magnitude lower samples compared to
+SG methods, and enables an order of magnitude faster
+inference compared to SS methods. Thus, the proposed
+model improves deep MRI reconstruction with elevated
+learning and computational efficiency.
+Index Terms— image reconstruction, deep learning, scan
+specific, scan general, low data, supervised, unsupervised.
+This work was supported in part by a TUBA GEBIP 2015 fellowship,
+by a BAGEP 2017 fellowship, and by a TUBITAK 121E488 grant awarded
+to T. C¸ ukur.
+S. UH. Dar, S¸ .
+¨Ozt¨urk, M.
+¨Ozbey, and T. C¸ ukur are with the
+Department of Electrical and Electronics Engineering, and the Na-
+tional Magnetic Resonance Research Center, Bilkent University,
+Ankara, Turkey (e-mails: {salman,muzaffer,cukur}@ee.bilkent.edu.tr,
+[email protected]). S¸ .
+¨Ozt¨urk is also with the Amasya
+University, Amasya, Turkey.
+I. INTRODUCTION
+The unparalleled soft-tissue contrast and non-invasiveness
+of MRI render it a preferred modality in many diagnostic
+applications [1], [2], and downstream imaging tasks such
+as classification [3] and segmentation [4], [5]. However, the
+adverse effects of low spin polarization at mainstream field
+strengths on the signal-to-noise ratio make it slower against
+alternate modalities such as CT [6]. Since long scan durations
+inevitably constrain clinical utility, there is an ever-growing
+interest in accelerated MRI methods to improve scan efficiency.
+Accelerated MRI involves an ill-posed inverse problem with the
+aim of mapping undersampled acquisitions in k-space to high-
+quality images corresponding to fully-sampled acquisitions.
+Conventional frameworks for solving this problem rely on
+parallel imaging (PI) capabilities of receive coil arrays [7],
+[8], in conjunction with hand-constructed MRI priors [9], [10].
+A joint objective is iteratively optimized comprising a data-
+consistency (DC) term based on the physical signal model,
+and a regularization term that enforces the MRI prior [9]. The
+physical model constrains reconstructed data to be consistent
+with acquired data while considering coil sensitivities and
+undersampling patterns [11]. Meanwhile, the regularization
+term, often based on a linear transform where data are assumed
+to be compressible [9], introduces suboptimality when the
+distribution of MRI data diverges from the hand-constructed
+prior.
+Deep learning (DL) methods have recently been adopted as
+a promising framework to improve reconstruction performance
+[12]–[16]. Inspired by traditional methods, a powerful approach
+is based on scan-specific (SS) priors that leverage the physical
+signal model to learn a reconstruction specific to each test
+scan, i.e. undersampled k-space data from a given test subject.
+Similar to autocalibration procedures in PI, a first group of
+SS methods perform training using a fully-sampled calibration
+region and then exercise learned dependencies in broader k-
+space [15]–[18]. Following the deep image prior technique, a
+second group of methods use unconditional CNNs as a native
+MRI prior [19]–[21]. These CNNs map low-dimensional latent
+variables onto MR images, and latents and network weights
+are optimized to ensure consistency to acquired data based on
+the physical signal model. In general, SS priors learned on
+each subject at test time avoid the need for separate training
+datasets, and promise improved reliability against atypical
+anatomy. However, they suffer from long inference times that
+can be prohibitive particularly when nonlinear networks are
+arXiv:2301.02613v1  [eess.IV]  6 Jan 2023
+
+EMB
+NPS
+UFFC
+SignalProcessing Society
+0222
+adopted [22]–[24].
+A fundamental alternative is to employ scan-general (SG)
+priors based on deep nonlinear networks that capture latent fea-
+tures of MR images [12]–[14], [25]–[33]. Numerous successful
+architectures have been reported including perceptrons [34],
+basic convolutional neural networks (CNNs) [35]–[38], residual
+or recurrent CNNs [29], [39]–[41], generative adversarial
+networks (GANs) [42]–[46], transformers [47], [48] and
+diffusion models [49], [50]. Physics-guided unrolled methods
+have received particular attention that combine the physical
+signal model as in traditional frameworks and regularization
+via a deep network serving as an SG prior [13], [27], [51]–[53].
+Reconstruction is achieved via serially alternated projections
+through the physical signal model and the SG prior [38], [40],
+[54]–[56]. However, under low-data regimes, the suboptimally
+trained SG prior introduces errors that are propagated across
+the unrolled architecture, compromising performance [6], [57],
+[58]. Furthermore, learning of SG priors requires large training
+datasets from several tens to hundreds of subjects [28], [59],
+[60], which can limit practicality.
+Here, we propose a novel parallel-stream fusion model
+(PSFNet) that consolidates SS and SG priors to enable data-
+efficient training and computation-efficient inference in deep
+MRI reconstruction1. PSFNet leverages an SS stream to perform
+linear reconstruction based on the physical signal model, and an
+SG stream to perform nonlinear reconstruction based on a deep
+network. Unlike conventional unrolled methods based on serial
+projections, here we propose a parallel-stream architecture with
+learnable fusion of SS and SG priors. Fusion parameters are
+adapted across cascades and training iterations to emphasize
+task-critical information. Comprehensive experiments on brain
+MRI datasets are reported to demonstrate PSFNet under both
+supervised and unsupervised settings [62]–[66]. PSFNet is
+compared against an unrolled SG method [27], two SS methods
+[17], [67], and conventional SPIRiT reconstructions [11].
+Compared to the unrolled model, PSFNet lowers training data
+requirements an order of magnitude. Compared to SS models,
+PSFNet offers significantly faster inference times. Our main
+contributions are summarized below:
+• A novel cascaded network architecture is introduced
+that adaptively fuses SS and SG priors across cascades
+and training iterations to improve learning-based MRI
+reconstruction in low-data regimes.
+• The SS prior facilitates learning of the SG prior with
+limited data, and empowers PSFNet to successfully
+generalize to out-of-domain samples.
+• The SG prior improves performance by capturing nonlin-
+ear residuals, and enhances resilience against suboptimal
+hyperparameter selection in the SS component.
+• Parallel-stream fusion of SS and SG priors yields robust
+performance with limited training data in both supervised
+and unsupervised settings.
+II. THEORY
+1see [61] for a preliminary version of this work presented at ISMRM 2021.
+A. Image Reconstruction in Accelerated MRI
+MRI reconstruction is an inverse problem that aims to recover
+an image from a respective undersampled acquisition:
+MFx = y
+(1)
+where F is the Fourier transform, M is the sampling mask
+defining acquired k-space locations, x is the multi-coil image to
+be reconstructed and y are acquired multi-coil k-space data. To
+improve problem conditioning, additional prior information re-
+garding the expected distribution of MR images is incorporated
+in the form of a regularization term:
+ˆx = arg min
+x
+λ||MFx − y||2
+2 + R(x)
+(2)
+where the first term enforces DC between reconstructed and
+acquired k-space data, R(x) reflects the MRI prior, and λ
+controls the balance between the DC and regularization terms.
+The DC term can be implemented by injecting the acquired
+values of k-space data into the reconstruction [13]. Thus,
+mapping through a DC block is given as:
+fDC(x) = F −1ΛFx +
+λ
+1 + λF −1y
+(3)
+where Λ is a diagonal matrix with diagonal entries set to
+1
+1+λ at
+acquired k-space locations and set to 1 in unacquired locations.
+In traditional methods, the regularization term is based on a
+hand-constructed transform domain where data are assumed to
+have a sparse representation [9]. For improved conformation to
+the distribution of MRI data, recent frameworks instead adopt
+deep network models to capture either SG priors learned from
+a large MRI database with hundreds of subjects, or SS priors
+learned from individual test scans. Learning procedures for the
+two types of priors are discussed below.
+SG priors: In MRI, SG priors are typically adopted to
+suppress aliasing artifacts in the zero-filled reconstruction (i.e.,
+inverse Fourier transform) of undersampled k-space acquisitions
+[27]. A deep network model that performs de-aliasing can be
+learned from a large training dataset of undersampled and
+corresponding fully-sampled k-space acquisitions, and then
+employed to implement R(.) in Eq. 2 during inference. The
+regularization term based on an SG prior is given as:
+RSG (x) = arg min
+x
+||CSG(F −1y; ˆθSG) − x||2
+2
+(4)
+where CSG is an image-domain deep network with learned
+parameters ˆθSG. The formulation in Eq. 4 assumes that CSG
+recovers multi-coil output images provided multi-coil input
+images. The parameters θSG for CSG can be learned based
+on a pixel-wise loss between reconstructed and ground-truth
+images. Training is conducted offline via an empirical risk
+minimization approach based on Monte Carlo sampling [13]:
+LSG(θSG) =
+N
+�
+n=1
+||CSG(F −1yn; θSG) − ˘xn||p
+(5)
+where N is the number of training scans, n is the training scan
+index, ||.||p denotes ℓp norm, ˘xn is the ground-truth multi-coil
+image derived from the fully-sampled acquisition for the nth
+scan, and yn are respective undersampled k-space data.
+
+3
+A common approach to build CSG is based on unrolled
+architectures that perform cascaded projections through CNN
+blocks to regularize the image and DC blocks to ensure
+conformance to the physical signal model [27]. Given a total
+of K cascades with tied CNN parameters across cascades, the
+mapping through the kth cascade is [13], [68], [69]:
+xr
+k = fDC
+�
+fSG
+�
+xr
+k−1; θSG
+��
+(6)
+where xr
+k is the image for the rth scan (that could be a training
+or test scan) at the output of the kth cascade (k ∈ [1, 2, ..., K]),
+and xr
+0 = F −1yr where yr are the acquired undersampled data
+for the rth scan. Meanwhile, fSG is the CNN block embedded
+in the kth cascade with parameters θSG.
+As the parameters of SG priors are trained offline and then
+frozen during inference, deeper network architectures can be
+used for enhanced reconstruction performance along with fast
+inference. However, learning deep networks requires substantial
+training datasets that may be difficult to collect. Moreover,
+since SG priors learn aggregate representations of MRI data
+across training subjects, they may show poor generalization to
+subject-specific variability in anatomy [19].
+SS priors: Unlike SG priors, SS priors are not learned from
+a dedicated training dataset but instead they are learned directly
+for individual test scans to improve generalization [15]. The
+SS prior can also be used to implement R(.) in Eq. 2 with the
+respective regularization term expressed as:
+RSS (x) = arg min
+x
+||CSS(F −1y; ˆθSS) − x||2
+2
+(7)
+where CSS is an image-domain network with parameters ˆθSS.
+In the absence of ground-truth images, the parameters θq
+SS for
+the qth test scan can be learned based on proxy k-space losses
+between reconstructed and acquired undersampled data [22].
+Learning is conducted online to minimize this proxy loss:
+LSS(θq
+SS) = ||MFCSS(F −1yq; θq
+SS) − yq||p
+(8)
+where yq are acquired undersampled k-space data for the qth
+scan. An unrolled architecture can be adopted to build CSS
+by performing cascaded projections through network and DC
+blocks, resulting in the following mapping for the kth cascade:
+xq
+k = fDC
+�
+fSS
+�
+xq
+k−1; θq
+SS
+��
+(9)
+fSS can be operationalized as a linear or nonlinear network [22],
+[23]. As the parameters of SS priors are learned independently
+for each test scan, they promise enhanced generalization to
+subject-specific anatomy. However, since training is performed
+online during inference, SS priors can introduce substantial
+computational burden, particularly when deep nonlinear net-
+works are used that also increase the risk of overfitting [70].
+B. PSFNet
+Here, we propose to combine SS and SG priors to maintain
+a favorable trade-off between generalization performance
+and computational efficiency under low-data regimes. In the
+conventional unrolling framework, this requires computation
+of serially alternated projections through the SS, SG and DC
+blocks:
+xr
+k = fDC
+�
+fSG
+�
+fSS
+�
+xr
+k−1; θr
+SS
+�
+; θSG
+��
+(10)
+The unrolled architecture with K cascades can be learned
+offline using the training set. Note that scarcely-trained SG
+blocks under low-data regimes can perform suboptimally,
+introducing residual errors in their output. In turn, these errors
+will accumulate across serial projections to degrade the overall
+performance.
+To address this limitation, here we introduce a novel
+architecture, PSFNet, that performs parallel-stream fusion of
+SS and SG priors as opposed to the serial combination in
+conventional unrolled methods. PSFNet utilizes a nonlinear SG
+prior for high performance, and a linear SS prior to enhance
+generalization without excessive computational burden. The
+two priors undergo parallel-stream fusion with learnable fusion
+parameters η and γ, as displayed in Figure 1. These parame-
+ters adaptively control the relative weighting of information
+extracted by the SG versus SS streams during the course of
+training in order to alleviate error accumulation. As such, the
+mapping through the kth cascade in PSFNet is:
+xr
+k = ηkfDC(fSS(xr
+k−1; θr
+SS))+γkfDC(fSG(xr
+k−1; θSG))
+(11)
+In Eq. 11, the learnable fusion parameters for the SS and
+SG blocks at the kth cascade are ηk and γk, respectively. To
+enforce fidelity to acquired data, DC projections are performed
+on the outputs of SG and SS blocks. In PSFNet, the SG prior
+is learned collectively from the set of training scans and then
+frozen during inference on test scans. In contrast, the SS prior
+is learned individually for each scan, during both training and
+inference.
+Training: PSFNet involves a training phase to learn model
+parameters for the SG prior as well as its fusion with the SS
+prior. For each individual scan in the training set, PSFNet
+learns a dedicated SS prior for the given scan. Since learning
+of a nonlinear SS prior has substantial computational burden,
+we adopt a linear SS prior in PSFNet. In particular, the SS
+block performs dealiasing via convolution with a linear kernel
+[71]:
+fSS(xn
+k−1; θn
+SS) = F −1{θn
+SS ⊛ Fxn
+k−1}
+(12)
+where θn
+SS ∈ C(z×z×w×w) with n denoting the training scan
+index, z denoting the number of coil elements, and w denoting
+the kernel size in k-space. The SS blocks contain unlearned
+Fourier and inverse Fourier transformation layers as their input
+and output layers, respectively, and convolution is computed
+over the spatial frequency dimensions in k-space. Meanwhile,
+the SG prior is implemented as a deep CNN operating in image
+domain:
+fSG(xn
+k−1; θSG) = CNN(xn
+k−1)
+(13)
+Across the scans in the training set, the training loss for PSFNet
+
+4
+Fig. 1: (a) PSFNet comprises a parallel-stream cascade of sub-networks where each sub-network contains (b) a scan-general
+(SG) block, and (c) a scan-specific (SS) block. The two parallel blocks are each succeeded by (d) a data-consistency (DC)
+block, and their outputs are aggregated with learnable fusion weights, ηk and γk where k is the cascade index. At the end of K
+cascades, coil-combination is performed on multi-coil data using sensitivity maps estimated via ESPIRiT [71]. The SG block is
+implemented as a deep convolutional neural network (CNN) and the SS block was implemented as a linear projection layer.
+can then be expressed in constrained form as:
+LP SF Net(θSG,γγγ,ηηη) =
+N
+�
+n=1
+||ηKfDC(fSS(xn
+K−1; ˆθn
+SS))
++ γKfDC(fSG(xn
+K−1; θSG)) − ˘xn||p
+s.t. ˆθn
+SS = arg min
+θn
+SS
+||F −1W nyn − fSS(F −1W nyn; θn
+SS)||2
+2
+(14)
+The constraint in Eq. 14 corresponds to the scan-specific
+learning of the SS prior ˆθn
+SS, which is then adopted to
+calculate the loss. Assuming that the linear relationships among
+neighboring spatial frequencies are similarly distributed across
+k-space [71], ˆθn
+SS is learned by solving a self-regression
+problem on the subset of fully-sampled data in central k-space,
+where W n is a mask operator that selects data within this
+calibration region.
+Note that, unlike deep reconstruction models purely based
+on SG priors, the SG prior in PSFNet is not directly trained to
+remove artifacts in zero-filled reconstructions of undersampled
+data. Instead, the SG prior is trained to concurrently suppress
+artifacts in reconstructed images along with the SS prior; and
+the relative importance attributed to the two priors is determined
+by the fusion parameters at each cascade. As such, the SS prior
+can be given higher weight during initial training iterations
+where the SG prior is scarcely trained, whereas its weight can
+be relatively reduced during later iterations once the SG prior
+has been sufficiently trained. This adaptive fusion approach
+thereby lowers reliance on the availability of large training
+sets.
+Inference: During inference on the qth test scan, the respec-
+tive SS prior is learned online as:
+ˆθq
+SS = arg min
+θq
+SS
+||F −1W qyq − f q
+SS(F −1W qyq; θq
+SS)||2
+2
+(15)
+Afterwards, the learned ˆθq
+SS is used along with the previously
+trained ˆθSG to perform repeated projections through K cas-
+cades as described in Eq. 11. The multi-coil image recovered
+by PSFNet at the output of the K cascade is:
+ˆxq = ηKfDC(fSS(xq
+K−1; ˆθq
+SS))+γKfDC(fSG(xq
+K−1; ˆθSG))
+(16)
+where ˆxq denotes the recovered image. The final reconstruction
+can be obtained by performing combination across coils:
+ˆxq
+combined = A∗ˆxq
+(17)
+where A are coil sensitivities, and A∗ denotes the conjugate
+of A.
+III. METHODS
+A. Implementation Details
+In each cascade, PSFNet contained two parallel streams
+with SG and SS blocks. The SG blocks comprised an input
+layer followed by a stack of 4 convolutional layers with 64
+channels and 3x3 kernel size each, and an output layer with
+ReLU activation functions. They processed complex images
+with separate channels for real and imaginary components. The
+SS blocks comprised a Fourier layer, 5 projection layers with
+identity activation functions, and an inverse Fourier layer. They
+processed complex images directly without splitting real and
+imaginary components. The linear convolution kernel used in
+the projection layers was learned from the calibration region
+by solving a Tikhonov regularized self-regression problem [11].
+The DC blocks comprised 3 layers respectively to implement
+
+a) Parallel-Stream Fusion Model (PSFNet)
+Reconstructed
+Zero-Filled
+Nk
+Reconstruction
+Image
+Coil
+Combination
+Z
+Z
+: b) Scan-General
+c) Scan-Specific
+d) Data-Consistency:
+ReLU
+ReLU
+Conv
+ReLU
+Conv
+ReLU
+Conv
+LU
+Conv
+Conv
+Rel
+ce
+oni
+eLU
+ReLU
+Conv
+Conv
+Conv
+Conv
+Conv
+Xout-im
+ReLi
+ReLi
+ReL!
+R
+Yin5
+forward Fourier transformation, restoration of acquired k-
+space data and inverse Fourier transformation. PSFNet was
+implemented with 5 cascades, K=5. The weights of SG, SS, and
+DC blocks were tied across cascades to limit model complexity
+[27]. The only exception were fusion coefficients that determine
+the relative weighting of the SG and SS blocks at each
+stage (γ1, .., γk, ..., γ5 η1, ...ηk, ..., η5). These fusion parameters
+were kept distinct across cascades. Coil-combination on the
+recovered multi-coil images was performed using sensitivity
+maps estimated via ESPIRiT [71].
+B. MRI Dataset
+Experimental demonstrations were performed using brain
+MRI scans from the NYU fastMRI database [72]. Here, contrast-
+enhanced T1-weighted (cT1-weighted) and T2-weighted acquisi-
+tions were considered. The fastMRI dataset contains volumetric
+MRI data with varying image and coil dimensionality across
+subjects. Note that a central aim of this work was to sys-
+tematically examine the learning capabilities of models for
+varying number of training samples. To minimize potential
+biases due to across-subject variability in MRI protocols, here
+we selected subjects with matching imaging matrix size and
+number of coils. To do this, we only selected subjects with at
+least 10 cross-sections and only the central 10 cross-sections
+were retained in each subject. We further selected subjects
+with an in-plane matrix size of 256x320 for cT1 acquisitions,
+and of 288x384 for T2 acquisitions. Background regions in
+MRI data with higher dimensions were cropped. Lastly, we
+restricted our sample selection to subjects with at least 5 coil
+elements, and geometric coil compression [73] was applied to
+unify the number of coils to 5 in all subjects.
+Fully-sampled acquisitions were retrospectively undersam-
+pled to achieve acceleration rates of R=4x and 8x. Random
+undersampling patterns were designed via either a bi-variate
+normal density function peaking at the center of k-space, or a
+uniform density function across k-space. The standard deviation
+of the normal density function was adjusted to maintain
+the expected value of R across k-space. The fully-sampled
+calibration region spanned a 40x40 window in central k-space.
+C. Competing Methods
+PSFNet was compared against several state-of-the-art ap-
+proaches including SG methods, SS methods, and traditional
+PI reconstructions. For methods containing SG priors, both
+supervised and unsupervised variants were implemented.
+PSFNet: A supervised variant of PSFNet was trained using
+paired sets of undersampled and fully-sampled acquisitions.
+PSFNetUS: An unsupervised variant of PSFNet was im-
+plemented using self-supervision based on only undersampled
+training data. Acquired data were split into two non-overlapping
+sets where 40% of samples was reserved for evaluating the
+training loss and 60% of samples was reserved to enforce DC
+[64].
+MoDL: A supervised SG methods based on an unrolled
+architecture with tied weights across cascades was used [27].
+MoDL serially interleaves SG and DC blocks. The number of
+cascades and the structure of SG and DC blocks were identical
+to those in PSFNet.
+MoDLUS: An unsupervised variant of MoDL was imple-
+mented using self-supervision. A 40%-60% split was performed
+on acquired data to evaluate the training loss and enforce data
+consistency, respectively [64].
+sRAKI-RNN: An SS method was implemented based on the
+MoDL architecture [67]. Learning was performed to minimize
+DC loss on the fully-sampled calibration region. Calibration
+data were randomly split with 75% of samples used to define
+the training loss and 25% of samples reserved to enforce DC.
+Multiple input-output pairs were produced for a single test
+sample by utilizing this split.
+SPIRiT: A traditional PI reconstruction was performed using
+the SPIRiT method [11]. Reconstruction parameters including
+the regularization weight for kernel estimation (κ), kernel size
+(w), and the number of iterations (Niter) were independently
+optimized for each reconstruction task via cross-validation.
+SPARK: An SS method was used to correct residual errors
+from an initial SPIRiT reconstruction [17]. Learning was
+performed to minimize DC loss on the calibration region. The
+learned SS prior was then used to correct residual errors in
+the remainder of k-space.
+D. Optimization Procedures
+For all methods, hyperparameter selection was performed
+via cross-validation on a three-way split of data across subjects.
+There was no overlap among training, validation and test sets
+in terms of subjects. Data from 10 subjects were reserved for
+validation, and data from a separate set of 40 subjects were
+reserved for testing. The number of subjects in the training
+set was varied from 1 to 50. Hyperparameters that maximized
+peak signal-to-noise ratio (PSNR) on the validation set were
+selected for each method.
+Training was performed via the Adam optimizer with
+learning rate ζ=10−4, β1=0.90 and β2=0.99 [74]. All deep
+learning methods were trained to minimize hybrid ℓ1-ℓ2-
+norm loss between recovered and target data (e.g., between
+reconstructed and ground truth images for PSFNet, between
+recovered and acquired k-space samples for PSFNetUS) [64].
+For PSFNet and MoDL, the selected number of epochs was
+200, batch size was set to 2 for the limited number of training
+samples (Nsamples <10), and to 5 otherwise. In DC blocks,
+λ = ∞ was used to enforce strict data consistency. For PSFNet
+and SPIRiT, the kernel width (w) and regularization parameter
+(κ) values were set as (κ, w) = (10−2, 9) at R= 4 and (10−2,
+9) at R=8 for cT1-weighted reconstructions, and as (100, 17)
+at R=4 and (10−2, 17) at R=8 for T2-weighted reconstructions.
+For SPIRiT, the number of iterations Niter was set as 13
+at R=4 and 27 at R=8 for cT1-weighted reconstructions, 20
+at R=4 and 38 at R=8 for T2-weighted reconstructions. For
+sRAKI-RNN, the selected number of epochs was 500 and
+batch size was set to 32. All other optimization procedures
+were identical to MoDL. For SPARK, network architecture
+and training procedures were adopted from [17], except for the
+number of epochs (Nepoch) and learning rate (ζ) which were
+optimized on the validation set as (Nepoch, ζ)= (100, 10−2) For
+
+6
+Fig. 2: Average PSNR across test subjects for (a) cT1- and (b)
+T2-weighted image reconstructions at R=4x. Model training was
+performed for varying number of training samples (Nsamples,
+lower x-axis) and thereby training subjects (Nsubjects, upper
+x-axis). Results are shown for SPIRiT, SPARK, sRAKI-RNN,
+MoDL and PSFNet.
+cT1-weighted reconstructions, and (Nepoch, ζ)= (250, 10−3)
+for T2-weighted reconstructions.
+All competing methods were executed on an NVidia RTX
+3090 GPU, and models were coded in Tensorflow except for
+SPARK which was implemented in PyTorch. SPARK was
+implemented using the toolbox at https://github.com/
+YaminArefeen/spark_mrm_2021. The code to imple-
+ment PSFNet will be available publicly at https://github.
+com/icon-lab/PSFNet upon publication.
+E. Performance Metrics
+Performance assessments for reconstruction methods were
+carried out by visual observations and quantitative metrics.
+PSNR and structural similarity index (SSIM) were used
+for quantitative evaluation. For each method, metrics were
+computed on coil-combined images from the reconstruction
+and from the fully-sampled ground truth acquisition. Statistical
+differences between competing methods were examined via
+non-parametric Wilcoxon signed-rank tests.
+F. Experiments
+Several different experiments were conducted to system-
+atically examine the performance of competing methods.
+Assessments aimed to investigate reconstruction performance
+under low training data regimes, generalization performance
+in case of mismatch between training and testing domains,
+contribution of the parallel-stream design to reconstruction per-
+formance, sensitivity to hyperparameter selection, performance
+in unsupervised learning, and computational complexity.
+Performance in low-data regimes: Deep SG methods for
+MRI reconstruction typically suffer from suboptimal perfor-
+mance as the size of the training dataset is constrained. To
+systematically examine reconstruction performance, we trained
+supervised variants of PSFNet and MoDL while the number
+of training samples (Nsamples) was varied in the range [2-
+500] cross sections. To attain a given number of samples,
+sequential selection was performed across subjects and across
+cross-sections within each subject. Thus, the number of unique
+subjects included in the training set roughly corresponded to
+Nsamples/10 (since there were 10 cross-sections per subject).
+SS reconstructions were also performed with sRAKI-RNN,
+SPIRiT and SPARK. In the absence of fully-sampled ground
+truth data to guide the learning of the prior, unsupervised
+training of deep reconstruction models may prove relatively
+more difficult compared to supervised training. In turn, this
+may elevate requirements on training datasets for unsupervised
+models. To examine data efficiency for unsupervised training,
+we compared the reconstruction performance of PSFNetUS and
+MoDLUS as Nsamples was varied in the range of [2-500] cross
+sections. Comparisons were also provided against sRAKI-RNN,
+SPIRiT and SPARK.
+Generalization performance: Deep reconstruction models
+can suffer from suboptimal generalization when the MRI data
+distribution shows substantial variation between the training and
+testing domains. To examine generalizability, PSFNet models
+were trained on data from a source domain and tested on data
+from a different target domain. The domain-transferred models
+were then compared to models trained and tested directly
+in the target domain. Three different factors were altered to
+induce domain variation: tissue contrast, undersampling pattern,
+and acceleration rate. First, the capability to generalize to
+different tissue contrasts was evaluated. Models were trained
+on data from a source contrast and tested on data from
+a different target contrast. Domain-transferred models were
+compared to target-domain models trained on data from the
+target contrast. Next, the capability to generalize to different
+undersampling patterns was assessed. Models were trained on
+data undersampled with variable-density patterns and tested
+on data undersampled with uniform-density patterns. Domain-
+transferred models were compared to target-domain models
+trained on uniformly undersampled data. Lastly, the capability
+to generalize to different acceleration rates was examined.
+Models were trained on acquisitions accelerated at R=4x and
+tested on acquisitions accelerated at R=8x. Domain-transferred
+models were compared to target-domain models trained at
+R=8x.
+Sensitivity to hyperparameters: SS priors are learned
+from individual test scans as opposed to SG priors trained
+on larger training datasets. Thus, SS priors might show
+elevated sensitivity to hyperparameter selection. We assessed
+the reliability of reconstruction performance against suboptimal
+
+Nsubjects
+(r
+X
+40
+39
+PSNR (dB)
+38
+SPIRiT
+37
+SPARK
+36
+SRAKI-RNN
+MoDL
+35
+PSFNet
+34
+Nsamples
+b)
+Nsubjects
+X
+6
+40
+39
+38
+PSNR (dB)
+SPIRiT
+SPARK
+SRAKI-RNN
+MoDL
+34
+PSFNet
+33
+Nsamples7
+Fig. 3: cT1-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDL, and PSFNet along with the
+zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each method
+are shown in the bottom row. MoDL and PSFNet were trained on 10 cross-sections from a single subject. SPIRiT, SPARK and
+sRAKI-RNN directly performed inference on test data without a priori model training. PSFNet shows superior performance to
+competing methods in terms of residual reconstruction errors.
+Fig. 4: T2-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDL, and PSFNet along with the
+zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each method
+are shown in the bottom row. MoDL and PSFNet were trained on 10 cross-sections from a single subject. SPIRiT, SPARK and
+sRAKI-RNN directly performed inference on test data without a priori model training. PSFNet shows superior performance to
+competing methods in terms of residual reconstruction errors.
+hyperparameter selection for SS priors. For this purpose,
+analyses were conducted on SPIRiT, SPARK and PSFNet
+that embody SS methods to perform linear reconstructions
+in k-space. The set of hyperparameters examined included
+regularization parameters for kernel estimation (κ) and kernel
+size (w). Separate models were trained using κ in range [10-3-
+100] and w in range [5-17].
+Computational complexity: Finally, we assessed the com-
+putational complexity of competing methods. For each method,
+training and inference times were measured for a single subject
+with 10 cross-sections. Each cross-section had an imaging
+matrix size of 256x320 and contained data from 5 coils. For
+all methods including SS priors, hyperparameters optimized
+for cT1-weighted reconstructions at R=4 were used.
+Ablation analysis: To assess the contribution of the parallel-
+stream design in PSFNet, a conventional unrolled variant
+of PSFNet was formed, named as PSFNetSerial. PSFNetSerial
+combined the SG and SS priors via serial projections as
+described in Eq. 10. Modeling procedures and the design of
+SG and SS blocks were kept identical between PSFNet and
+PSFNetSerial for fair comparison. Performance was assessed as
+Nsamples was varied in the range of [2-500] cross sections.
+IV. RESULTS
+A. Performance in Low-Data Regimes
+Common SG methods for MRI reconstruction are based
+on deep networks that require copious amounts of training
+data, so performance can substantially decline on limited
+training sets [28], [59]. In contrast, PSFNet leverages an SG
+prior to concurrently reconstruct an image along with an SS
+prior. Therefore, we reasoned that its performance should scale
+favorably under low-data regimes compared to SG methods. We
+also reasoned that PSFNet should yield elevated performance
+compared to SS methods due to residual corrections from its SG
+prior. To test these predictions, we trained supervised variants
+of PSFNet and MoDL along with SPIRiT, sRAKI-RNN, and
+SPARK while the number of training samples (Nsamples) was
+
+ZF
+SPIRiT
+SPARK
+SRAKI-RNN
+MoDI
+PSFNet
+Reference
+0.12
+Error
+0ZF
+SPIRiT
+SPARK
+SRAKI-RNN
+MoDL
+PSFNet
+Reference
+0.12
+Error
+08
+Fig. 5: Weighting of the SG (γ) and SS (η) blocks in the
+final cascade of PSFNet. Weights were averaged across models
+trained for cT1- and T2-weighted reconstructions at R=4x.
+Model training was performed for varying number of training
+samples (Nsamples, lower x-axis) and thereby training subjects
+(Nsubjects, upper x-axis). Both blocks are equally weighted
+with very limited training data. As Nsamples increases, the
+weighting of the SG prior becomes more dominant over the
+weighting of the SS prior.
+systematically varied. Figure 2 displays PSNR performance
+for cT1-weighted and T2-weighted image reconstruction as a
+function of Nsamples. PSFNet outperforms the scan-general
+MoDL method for all values of Nsamples (p < 0.05). As
+expected, performance benefits with PSFNet become more
+prominent towards lower values of Nsamples. PSFNet also
+outperforms traditional SPIRiT and scan-specific sRAKI-RNN
+and SPARK methods broadly across the examined range
+of Nsamples (p < 0.05). Note that while MoDL requires
+Nsamples = 30 (3 subjects) to offer on par performance to
+SS methods, PSFNet yields superior performance with as few
+as Nsamples = 2. Representative reconstructions for cT1- and
+T2-weighted images are depicted in Figures 3 and 4, where
+Nsamples = 10 from a single subject were used for training.
+PSFNet yields lower reconstruction errors compared to all other
+methods in this low-data regime, where competing methods
+either show elevated noise or blurring.
+Naturally, the performance of PSFNet increases as more
+training samples are available. Since the SS prior is inde-
+pendently learned for individual samples, it should not elicit
+systematic performance variations depending on Nsamples.
+Thus, the performance gains can be attributed to improved
+learning of the SG prior. In turn, we predicted that PSFNet
+would put more emphasis on its SG stream as its reliability
+increases. To examine this issue, we inspected the weightings
+of the SG (γ) and SS (η) streams as the training set size was
+varied. Figure 5 displays weightings at the last cascade as a
+function of Nsamples. For lower values of Nsamples where the
+quality of the SG prior is relatively limited, the SG and SS
+priors are almost equally weighted. In contrast, as the learning
+of the SG prior improves with higher Nsamples, the emphasis
+on the SG prior increases while the SS prior is less heavily
+Fig. 6: Average PSNR across test subjects for (a) cT1- and (b)
+T2-weighted image reconstructions at R=4x. Model training was
+performed for varying number of training samples (Nsamples,
+lower x-axis) and thereby training subjects (Nsubjects, upper
+x-axis). Results are shown for SPIRiT, SPARK, sRAKI-RNN,
+MoDLUS and PSFNetUS.
+weighted.
+We then questioned whether the performance benefits of
+PSFNet are also apparent during unsupervised training of
+deep network models. For this purpose, unsupervised variants
+PSFNetUS and MoDLUS were trained via self-supervision [64].
+PSFNetUS was compared against MoDLUS, SPIRiT, sRAKI-
+RNN, and SPARK while the number of training samples
+(Nsamples) was systematically varied. Figure 6 displays PSNR
+performance for cT1-weighted and T2-weighted image recon-
+struction as a function of Nsamples. Similar to the supervised
+setting, PSFNetUS outperforms MoDLUS for all values of
+Nsamples (p < 0.05), and the performance benefits are more
+noticeable at lower Nsamples. In this case, however, MoDLUS
+is unable to reach the performance of the best performing
+SS method (SPARK) even at Nsamples = 500. In contrast,
+PSFNetUS starts outperforming SPARK with approximately
+Nsamples = 50 (5 subjects). The enhanced reconstruction
+quality with PSFNetUS is corroborated in representative re-
+constructions for cT1- and T2-weighted images depicted in
+Figures 7 and 8, where Nsamples = 100 were used for training.
+Taken together, these results indicate that the data-efficient
+nature of PSFNet facilitates the training of both supervised
+and unsupervised MRI reconstruction models.
+
+Nsubjects
+5
+6
+Y (SG)
+0.60
+n (Ss)
+0.55
+Weighting
+0.50
+0.45
+0.40
+4
+6
+8
+NsamplesNsubjects
+a
+X
+40
+39
+38
+37
+(dB)
+36
+SPIRiT
+35
+PSNR
+34
+SPARK
+SRAKI - RNN
+33
+MoDLus
+32
+PSFNetus
+31
+30
+2
+X
+Nsamples
+b)
+Nsubjects
+5
+6
+40
+39
+38
+37
+(dB)
+36
+.343
+SPIRiT
+PSNR 
+SPARK
+SRAKI - RNN
+MoDLus
+32
+PSFNetus
+31
+30
+X
+Nsamples9
+Fig. 7: cT1-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDLUS, and PSFNetUS along with
+the zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each
+method are shown in the bottom row. MoDLUS and PSFNetUS were trained on 100 cross-sections (from 10 subjects). SPIRiT,
+SPARK and sRAKI-RNN directly performed inference on test data without a priori model training. PSFNetUS shows superior
+performance to competing methods in terms of residual reconstruction errors.
+Fig. 8: T2-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDLUS, and PSFNetUS along with
+the zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each
+method are shown in the bottom row. MoDLUS and PSFNetUS were trained on 100 cross-sections (from 10 subjects). SPIRiT,
+SPARK and sRAKI-RNN directly performed inference on test data without a priori model training. PSFNetUS shows superior
+performance to competing methods in terms of residual reconstruction errors.
+B. Generalization Performance
+An important advantage of SS priors is that they allow
+model adaptation to individual test samples, thereby promise
+enhanced performance in out-of-domain reconstructions [22].
+Yet, SG priors with fixed parameters might show relatively
+limited generalizability during inference [23], [75]. To assess
+generalization performance, we introduced domain variations
+by altering three experimental factors: tissue contrast, under-
+sampling pattern, and acceleration rate. For methods comprising
+SG components, we built both target-domain models that were
+trained in the target domain, and domain-transferred models
+that were trained in a non-target domain. We then compared
+the reconstruction performances of the two models in the target
+domain.
+First, we examined generalization performance when the
+tissue contrast varied between training and testing domains
+(e.g., trained on cT1, tested on T2). Table I lists performance
+metrics for competing methods with Nsamples = 500. While
+performance losses are incurred for domain-transferred PSFNet-
+DT and MoDL-DT models that contain SG components, these
+losses are modest. On average, MoDL-DT shows a loss of
+0.3dB PSNR and 0.1% SSIM (p < 0.05), and PSFNet-DT
+shows a loss of 0.2dB PSNR and 0.1% SSIM (p < 0.05). Note
+that PSFNet-DT still outperforms the closest competing SS
+method by 2.2dB PSNR and 1.8% SSIM (p < 0.05).
+Second, we examined generalization performance when mod-
+els were trained with variable-density and tested on uniform-
+density undersampling patterns. Table II lists performance
+metrics for competing methods. On average across tissue
+contrasts, MoDL-DT suffers a notable performance loss of
+3.6dB PSNR and 2.5% SSIM (p < 0.05). In contrast, PSFNet-
+DT shows a relatively limited loss of 0.4dB PSNR and 0.2%
+SSIM (p < 0.05). Note that PSFNet-DT again outperforms the
+closest competing SS method by 3.4dB PSNR and 3.7% SSIM
+(p < 0.05).
+Third, we examined generalization performance when models
+were trained at R=4x and tested on R=8x. Table III lists
+
+ZF
+SPIRiT
+SPARK
+sRAKI-RNN
+MoDL
+PSFNet
+Reference
+0.12
+Error
+0ZF
+SPIRiT
+SPARK
+sRAKI-RNN
+MoDL
+PSFNet
+Reference
+0.12
+Error
+010
+TABLE I: Generalization across tissue contrasts. PSNR and
+SSIM values (mean±standard error) across test subjects.
+Results are shown for scan-specific models (SPIRiT, SPARK,
+sRAKI-RNN), target-domain models (MoDL, PSFNet) and
+domain-transferred models (MoDL-DT, PSFNet-DT) at R=4x.
+The tissue contrast in the target domain is listed in the left-most
+column (cT1 or T2), domain-transferred models were trained
+for the non-target tissue contrast.
+SPIRiT
+SPARK
+sRAKI-
+RNN
+MoDL
+MoDL-DT
+PSFNet
+PSFNet-
+DT
+PSNR
+cT1
+37.6
+37.6
+36.8
+38.5
+38.2
+39.9
+39.4
+±1.5
+±1.5
+±1.3
+± 1.5
+±1.5
+±1.7
+±1.6
+T2
+35.8
+36.5
+35.2
+37.9
+37.5
+39.0
+39.0
+±1.0
+±1.0
+±1.1
+± 1.0
+±1.1
+±1.0
+±0.9
+SSIM
+cT1
+93.1
+93.3
+93.8
+95.1
+94.8
+95.8
+95.6
+±1.5
+±1.4
+±1.0
+±1.0
+±1.1
+±1.0
+±1.0
+T2
+90.8
+93.1
+94.9
+96.2
+96.2
+96.7
+96.8
+±1.2
+±1.0
+±0.6
+±0.5
+±0.5
+±0.4
+±0.4
+TABLE II: Generalization across undersampling patterns. PSNR
+and SSIM values (mean±standard error) across test subjects.
+Results are shown for Results are shown for scan-specific
+models (SPIRiT, SPARK, sRAKI-RNN), target-domain models
+(MoDL, PSFNet) and domain-transferred models (MoDL-DT,
+PSFNet-DT) at R=4x. Domain-transferred models were trained
+with variable-density undersampling, and tested on uniform-
+density undersampling. Target-domain models were trained and
+tested with uniform-density undersampling.
+SPIRiT
+SPARK
+sRAKI-
+RNN
+MoDL
+MoDL-DT
+PSFNet
+PSFNet-
+DT
+PSNR
+cT1
+37.1
+37.1
+33.6
+37.0
+33.6
+40.2
+39.9
+±1.8
+±1.7
+±1.4
+± 1.7
+±1.8
+±1.6
+±1.6
+T2
+35.1
+35.6
+31.6
+37.0
+33.2
+40.2
+39.7
+±1.3
+±1.3
+±1.5
+± 1.1
+±1.2
+±1.1
+±1.2
+SSIM
+cT1
+92.9
+93.0
+91.2
+93.4
+91.2
+95.9
+95.6
+±1.5
+±1.5
+±1.5
+±1.3
+±2.0
+±1.2
+±1.2
+T2
+90.6
+92.1
+91.5
+95.6
+92.7
+97.1
+96.9
+±1.5
+±1.5
+±1.2
+±0.7
+±1.1
+±0.6
+±0.6
+performance metrics for competing methods. On average across
+tissue contrasts, MoDL-DT suffers a notable performance loss
+of 1.0dB PSNR and performs slightly better in SSIM by
+0.2%SSIM (p < 0.05), whereas PSFNet-DT shows a lower loss
+of 0.6dB PSNR (p < 0.05) and performs similarly in SSIM
+(p > 0.05). PSFNet-DT outperforms the closest competing SS
+method by 1.2dB PSNR and 1.9% SSIM (p < 0.05). Taken
+together, these results clearly suggest that the SS prior in
+PSFNet contributes to its improved generalization performance
+over the scan-general MoDL method, while the SG prior in
+PSFNet enables it to outperform competing SS methods.
+C. Sensitivity to Hyperparameters
+Parameters of deep networks that implement SS priors are to
+be learned from a single test sample, so the resultant models can
+show elevated sensitivity to the selection of hyperparameters
+compared to SG priors learned from a collection of training
+TABLE III: Generalization across acceleration rates. PSNR
+and SSIM values (mean±standard error) across test subjects.
+Results are shown for scan-specific models (SPIRiT, SPARK,
+sRAKI-RNN), target-domain models (MoDL, PSFNet) and
+domain-transferred models (MoDL-DT, PSFNet-DT). Domain-
+transferred models were trained at R=4x and tested at R=8x.
+Target-domain models were trained and tested at R=8x.
+SPIRiT
+SPARK
+sRAKI-
+RNN
+MoDL
+MoDL-DT
+PSFNet
+PSFNet-
+DT
+PSNR
+cT1
+34.7
+34.8
+34.3
+35.3
+34.5
+36.5
+36.2
+±1.5
+±1.5
+±1.5
+± 1.4
+±1.7
+±1.5
+±1.5
+T2
+33.6
+33.7
+32.6
+34.6
+33.4
+35.6
+34.6
+±1.0
+±1.0
+±0.9
+± 1.0
+±1.2
+±1.1
+±1.2
+SSIM
+cT1
+89.8
+90.8
+91.4
+92.1
+92.2
+93.3
+93.3
+±1.9
+±1.6
+±1.4
+±1.5
+±1.4
+±1.4
+±1.4
+T2
+89.0
+90.1
+92.7
+93.5
+93.7
+94.6
+94.5
+±1.3
+±1.1
+±0.9
+±0.8
+±0.8
+±0.7
+±0.7
+samples. Thus, we investigated the sensitivity of PSFNet to key
+hyperparameters of its SS prior. SPIRiT, SPARK and PSFNet
+methods all embody a linear k-space reconstruction, so the
+relevant hyperparameters are the regularization weight and
+width for the convolution kernel. Performance was evaluated for
+models were trained in the low-data regime (i.e., Nsamples =
+10, 1 subject) for varying hyperparameter values.
+Figure 9 displays PSNR measurements for SPIRiT, SPARK
+and PSFNet across κ in range (10-3-100). While the perfor-
+mance of SPIRiT and SPARK is notably influenced by κ,
+PSFNet is minimally affected by sub-optimal selection. On
+average across contrasts, the difference between the maximum
+and minimum PSNR values is 8.4dB for SPIRiT, 4.5dB for
+SPARK, and a lower 0.7dB for PSFNet. Note that PSFNet
+outperforms competing methods across the entire range of
+κ (p < 0.05). Figure 10 shows PSNR measurements for
+competing methods across w in range (5-17). In this case,
+all methods show relatively limited sensitivity to the selection
+of w. On average across contrasts, the difference between the
+maximum and minimum PSNR values is 1.5dB for SPIRiT,
+0.5dB for SPARK, and 0.2dB for PSFNet. Again, PSFNet
+outperforms competing methods across the entire range of
+w (p < 0.05). Overall, our results indicate that PSFNet
+yields improved reliability against sub-optimal hyperparameter
+selection than competing SS methods.
+D. Computational Complexity
+Next, we assessed the computational complexity of com-
+peting methods. Table IV lists the training times of methods
+with SG priors, MoDL and PSFNet. Note that the remaining
+SS based methods do not involve a pre-training step. As it
+involves learning of an SS prior on each training sample,
+PSFNet yields elevated training time compared to MoDL. In
+return, however, it offers enhanced generalization performance
+and data-efficient learning. Table IV also lists the inference
+times of SPIRiT, SPARK, sRAKI-RNN, MoDL and PSFNet.
+MoDL and PSFNet that employ SG priors with fixed weights
+during inference offer fast run times. In contrast, SPARK and
+
+11
+Fig. 9: PSNR measurements were performed on recovered cT1-
+and T2-weighted images at R=4x. Bar plots in blue color show
+average PSNR across κ ∈ 10-3-101 (i.e., the regularization
+parameter for kernel estimation). Error bars denote the 90%
+interval across κ. Bar plots in red color show PSNR for methods
+that do not depend on the value of κ.
+Fig. 10: PSNR measurements were performed on recovered
+cT1- and T2-weighted images at R=4x. Bar plots in blue color
+show the average PSNR across w ∈ 5-17 (i.e., the kernel size).
+Error bars denote the 90% interval across w. Bar plots in red
+color show PSNR for methods that do not depend on the value
+of w.
+sRAKI-RNN that involve SS priors learned on individual test
+samples have a high computational burden. Although PSFNet
+also embodies an SS prior, its uses a relatively lightweight
+linear prior as opposed to the nonlinear priors in competing
+SS methods. Therefore, PSFNet benefits from data-efficient
+learning while maintaining computationally-efficient inference.
+E. Ablation Analysis
+To demonstrate the value of the parallel-stream fusion
+strategy in PSFNet over conventional unrolling, PSFNet was
+compared against a variant model PSFNetSerial that combined
+SS and SG priors through serially alternated projections.
+Separate models were trained with number of training samples
+in the range Nsamples=[2-500]. Performance in cT1- and T2
+-weighted image reconstruction is displayed in Figure 11.
+PSFNet significantly improves reconstruction performance over
+PSFNetSerial across the entire range of Nsamples considered
+(p < 0.05), and the benefits grow stronger for smaller training
+sets. On average across contrasts for Nsamples < 10, PSFNet
+TABLE IV: Computational complexity of competing methods.
+Training and inference times for data from a single subject,
+with 10 cross-sections, imaging matrix size 256x320 and 5
+coils. Run times are listed for SPARK, sRAKI-RNN, MoDL,
+and PSFNet.
+SPIRiT
+SPARK
+sRAKI-RNN
+MoDL
+PSFNet
+Training(s)
+-
+-
+-
+132
+337
+Inference(s)
+0.85
+23.35
+285.00
+0.25
+1.13
+Fig. 11: Average (a) PSNR and (b) SSIM values for cT1- and
+T2-weighted image reconstructions at R=4x. Model training was
+performed for varying number of training samples (Nsamples,
+lower x-axis) and thereby training subjects (Nsubjects, upper
+x-axis). Results are shown for PSFNet and PSFNetSerial.
+outperforms PSFNetSerial by 1.8dB PSNR and 0.6% SSIM
+(p < 0.05). These results indicate that the parallel-stream
+fusion of SG and SS priors in PSFNet is superior to the serial
+projections in conventional unrolling.
+V. DISCUSSION AND CONCLUSION
+In this study, we introduced PSFNet for data-efficient training
+of deep reconstruction models in accelerated MRI. PSFNet
+synergistically fuses SS and SG priors in a parallel-stream
+architecture. The linear SS prior improves learning efficiency
+while mataining relatively low computational footprint, whereas
+the nonlinear SG prior enables improved reconstruction per-
+formance. For both supervised and unsupervised training
+setups, the resulting model substantially reduces dependence
+on the availability of large MRI datasets. Furthermore, it
+achieves competitive inference times to SG methods, and
+
+38
+(dB)
+36
+PSNR
+34
+32
+30
+MoDL
+SPIRiT
+SPARK
+PSFNet
+SRAKI - RNN38
+(dB)
+36
+PSNR
+34
+32
+30
+MoDL
+SPIRiT
+SPARK
+PSFNet
+SRAKI - RNNNsubjects
+a
+X
+40
+39
+PSNR (dB)
+38
+37
+PSFNet (cTi)
+36
+PSFNet (T2)
+PSFNetserial (cTi)
+35
+PSFNetserial (T2)
+34
+Nsamples
+b)
+Nsubjects
+5
+X
+6
+97
+96
+(%)
+SSIM
+95
+PSFNet (cTi)
+PSFNet (T2)
+94
+PSFNetserial (cTi)
+PSFNetserial (T2)
+93
+Nsamples12
+reliably generalizes across tissue contrasts, sampling patterns
+and acceleration rates.
+Several prominent approaches have been introduced in
+the literature to address the training requirements of deep
+models based on SG priors. One approach is to pre-train
+models on readily available datasets from a separate source
+domain and then to fine-tune on several tens of samples
+from the target domain [28], [59] or else perform SS fine-
+tuning [76]. This transfer learning approach relaxes the domain
+requirements for training datasets. However, the domain-
+transferred models might be suboptimal when training and
+testing data distributions are divergent. In such cases, additional
+training for domain-alignment might be necessary to mitigate
+performance losses. In contrast, PSFNet contains a SS prior that
+allows it to better generalize to out-of-domain data without
+further training. Another approach is to build unsupervised
+models to alleviate dependency on training datasets with paired
+undersampled, fully-sampled acquisitions. Model training can
+be performed either directly on undersampled acquisitions
+via self-supervision [64] or on unpaired sets of undersampled
+and fully-sampled acquisitions via cycle-consistent learning
+[77]. This approach can prove beneficial when fully-sampled
+acquisitions are costly to collect. Nonetheless, the resulting
+models still require relatively large datasets form tens of
+subjects during training [64]. Note that our experiments on
+self-supervised variants of PSFNet and MoDL suggest that
+unsupervised models can be more demanding for data than their
+supervised counterparts. Therefore, the data-efficiency benefits
+of PSFNet might be particularly useful for unsupervised deep
+MRI reconstruction.
+A fundamentally different framework to lower requirements
+on training datasets while offering improved generalizability
+is based on SS priors. In this case, learning can be performed
+directly on test data and models can be adapted to each scan
+[15], [17]. A group of studies have proposed SS methods based
+on relatively compact nonlinear models to facilitate learning
+during inference [15], [17], [18], [78]. However, because learn-
+ing is performed in central k-space, these methods implicitly
+assume that local relationships among spatial frequency samples
+are largely invariant across k-space. While the SS prior in
+PSFNet also rests on a similar assumption, the SG components
+helps correct residual errors that can be introduced due to this
+assumption. Another group of studies have alternatively adopted
+the deep image prior (DIP) approach to build SS methods
+[19], [20], [22], [23]. In DIP, unconditional deep network
+models that map latent variables onto images are used as
+native priors for MR images. The priors are learned by ensuring
+the consistency of reconstructed and acquired data across the
+entire k-space. Despite improved generalization, these relatively
+more complex models require increased inference times. In
+comparison, PSFNet provides faster inference since the weights
+for its SG prior are fixed, and its SS prior involves a compact
+linear operator that is easier to learn.
+Few independent studies on MRI have proposed approaches
+related to PSFNet by combining nonlinear and linear recon-
+structions [6], [17], [78]. Residual RAKI and SPARK methods
+initially perform a linear reconstruction, and then use an SS
+method to correct residual errors via minimizing a DC loss in
+the calibration region [17], [78]. As local relationships among
+data samples might vary across k-space, the learned SS priors
+might be suboptimal. Moreover, these methods perform online
+learning of nonlinear SS priors that introduces relatively high
+computational burden. In contrast, PSFNet incorporates an SG
+prior to help improve reliability against sub-optimalities in the
+SS prior, and uses a linear SS prior for efficiency. Another
+related method is GrappaNet that improves reconstruction per-
+formance by cascading GRAPPA and network-based nonlinear
+reconstruction steps [6]. While [6] intends to improve image
+quality, the main aim of our study is to improve practicality
+by lowering training data requirements of deep models, and
+improving domain generalizability without elevating inference
+times. Note that GrappaNet follows the conventional unrolling
+approach by performing serially alternated projections through
+linear and nonlinear reconstructions, which can lead to error
+propagation under low-data regimes [79]. In contrast, PSFNet
+maintains linear and nonlinear reconstructions as two parallel
+streams in its architecture, and learns to optimally fuse the
+information from the two streams.
+The proposed method can be improved along several lines
+of technical development. First, to improve the capture of high-
+frequency information by the SG prior, an adversarial loss
+term along with a discriminator subnetwork can be included
+in PSFNet [80]. It remains to be demonstrated whether the
+data-efficiency benefits of PSFNet are apparent for adversarial
+training setups. Second, nonlinear activation functions can
+be included in the SS stream to improve the expressiveness
+of the SS prior [78]. While learning of nonlinear priors can
+elevate inference complexity, generalization performance might
+be further improved. Third, the expressiveness of both SS
+and SG priors might be enhanced by incorporating attention
+mechanisms as proposed in recent transformer models [81].
+Fourth, using multimodal image fusion approaches can improve
+performance in case of having a repository with multimodal
+data [82], [83]. Lastly, the benefits of transfer learning and
+PSFNet can be aggregated by pre-training the SG prior on
+natural images to further lower requirements on training data.
+VI. ACKNOWLEDGMENTS
+This work was supported in part by a TUBA GEBIP 2015
+fellowship, by a BAGEP 2017 fellowship, and by a TUBITAK
+121E488 grant awarded to T. C¸ ukur.
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+

4NE0T4oBgHgl3EQfeQCD/content/tmp_files/2301.02388v1.pdf.txt

@@ -0,0 +1,717 @@
+Generating corneal panoramic images from contact specular microscope images
+Yusuke Nagira1†, Yuzuha Hara2, Satoru Hiwa2
+Naoki Okumura3, Noriko Koizumi3 and Tomoyuki Hiroyasu2
+1Graduate School of Life and Medical Sciences, Doshisha University, Japan
+2Department of Biomedical Sciences and Informatics, Doshisha University, Japan
+3Department of Biomedical Engineering, Faculty of Life and Medical Sciences, Doshisha University, Japan
+(Tel: +81-774-65-6020, E-mail: [email protected])
+Abstract: The contact specular microscope has a wider angle of view than that of the non-contact specular microscope but still
+cannot capture an image of the entire cornea. To obtain such an image, it is necessary to prepare film on the parts of the image
+captured sequentially and combine them to create a complete image. This study proposes a framework to automatically generate
+an entire corneal image from videos captured using a contact specular microscope. Relatively focused images were extracted
+from the videos and panoramic compositing was performed. If an entire image can be generated, it is possible to detect guttae
+from the image and examine the extent of their presence. The system was implemented and the effectiveness of the proposed
+framework was examined. The system was implemented using custom-made composite software, Image Composite Software
+(ICS, K.I. Technology Co., Ltd., Japan, internal algorithms not disclosed), and a supervised learning model using U-Net was
+used for guttae detection. Several images were correctly synthesized when the constructed system was applied to 94 different
+corneal videos obtained from Fuchs endothelial corneal dystrophy (FECD) mouse model. The implementation and application
+of the method to the data in this study confirmed its effectiveness. Owing to the minimal quantitative evaluation performed, such
+as accuracy with implementation, it may pose some limitations for future investigations.
+Keywords: U-Net, Semantic Segmentation, Fuchs Endothelial Corneal Dystrophy, Corneal Endothelial Cell
+1. INTRODUCTION
+Fuchs endothelial corneal dystrophy (FECD) is a bilat-
+eral disease, wherein corneal endothelial cells are unable to
+maintain their hexagonal shape. It is characterized by the
+accelerated loss of corneal endothelial cells with changes in
+Descemet’s membrane, resulting in the formation of an ex-
+tracellular matrix called guttae [1]. In the United States, it
+is estimated that 4% of people over 40 years of age are af-
+fected by the disease, occurring more commonly in women
+and more frequently in people in their 40s and 50s. Corneal
+endothelial pump function is lost as the disease progresses,
+causing corneal edema [2]. Presently, corneal transplanta-
+tion is the only reliable treatment, and FECD accounts for
+39% of all corneal transplants performed, making it the most
+common cause of corneal transplantation worldwide [3].
+Rho kinase inhibitors have been reported to promote cell
+proliferation and adhesion to substrates, inhibit corneal en-
+dothelial cell apoptosis, and promote wound healing. There-
+fore, using Rho kinase inhibitor eye drops is a potential
+novel treatment approach alternative to corneal transplanta-
+tion [4][5].
+In drug discovery research for FECD, the state of the
+corneal endothelium, such as the guttae, is observed before
+and after the drug use and evaluated based on the increase
+or decrease in the number of cells. Doctors and researchers
+widely use specular microscopes to observe the state of the
+corneal endothelium.
+However, the range of the micro-
+scope’s imaging capability is limited and the current practice
+estimates the state of the entire cornea from its center. The
+endothelial cell density (ECD) is essential for understanding
+† Yusuke Nagira is the presenter of this paper.
+the pathogenesis of FECD. However, ECD cannot be mea-
+sured accurately owing to the presence of guttae; hence, the
+number of cells is measured manually [6].
+Using a mouse model to demonstrate the pathogenesis of
+FECD, studies have been conducted on the segmentation of
+guttae using U-Net and the calculation of cell density in areas
+excluding guttae [7][8]. In previous studies on the panoramic
+synthesis of the corneal endothelium, focused images were
+extracted manually and stitched together. Panoramic com-
+positing of the entire cornea is yet to be performed because
+the images were localized panoramic images and did not rep-
+resent the entire cornea [9]. Conventional panorama synthe-
+sis software such as AutoStitch [10] does not consider the
+order of the input images used for synthesis. Instead, it ex-
+tracts the image features using Scale-Invariant Feature Trans-
+form (SIFT) [11] and stitches the matching features together.
+When pasting an image, it is deformed and scaled. For im-
+ages with similar characteristics, such as corneal endothelial
+cells, the position of the image to be pasted may be incor-
+rect or the shape of the cells may be deformed owing to the
+deformation of the image. In this case, accurate values can-
+not be obtained when calculating the area of the cells or the
+percentage of guttae. Therefore, the original image needed
+to have as minimal deformations as possible in the combined
+image. Alternatively, it is necessary to provide a mechanism
+to access the original version of the image of interest.
+This study proposes a framework for a system that gen-
+erates images of the entire corneal endothelium from videos
+obtained using a contact specular microscope. In the pro-
+posed framework, the focused images are extracted from the
+video images, feature extraction is performed, and the im-
+ages are synthesized. During synthesis, the system reduces
+arXiv:2301.02388v1  [eess.IV]  6 Jan 2023
+
+or deforms the original image to the minimum and adds a
+feature that allows access to the original image of the area
+of interest. This study examined the proposed framework by
+building a system using two implementation methods. Fur-
+ther, we added a function for automatically detecting guttae
+using U-Net, a type of Deep Learning. This study presents a
+proposed framework and an example of its implementation.
+Quantitative evaluation of whole corneal endothelial images
+and gut detection is insufficient, which poses a limitation that
+must be addressed in future studies.
+2. FRAMEWORK OF CORNEAL PANORAMIC
+IMAGE GENERATION FROM CONTACT
+SPECULAR MICROSCOPE IMAGES
+2.1. Overview of the proposed framework
+Fig.1 presents an overview of the proposed framework.
+An image of the entire cornea was generated from a video
+frame of the cornea captured using a contact-type specu-
+lar microscope. First, a focused still image was extracted
+from the target video image (dataset 1). Second, the features
+of still images were extracted. Third, the matching feature
+points were combined to create a panoramic image. Next,
+the panoramic image was divided into grid regions and the
+most focused image was selected from each region. Guttae
+were detected in the combined panoramic images using deep
+learning. Additionally, still images of the corneal endothe-
+lium containing the guttae and mask images exhibiting the
+location of the guttae were used for the model generation
+(dataset 2).
+2.2. Extraction of in-focus images
+A group of still images was extracted from the captured
+videos. The entire corneal endothelium was captured and
+converted to a single frame-by-frame image. If there were
+N frames in the video, N images were the output in total
+that were then divided into five groups in chronological order.
+The image with the highest in-focus evaluation index was
+selected from each group.
+2.3. Creating panoramic images
+The algorithm for generating a single panoramic image is
+accomplished through the following steps. First, characteris-
+tic points in the images were extracted. Subsequently, a curve
+connecting the characteristic points was obtained. Here, the
+optimal curve connecting the extracted characteristic points
+was obtained. This curve was then used to enlarge or interpo-
+late the image. A panoramic image was generated. In the fol-
+lowing experiments, two algorithms were prepared and the
+results were compared.
+2.4. Image sharpening process
+The synthesized panoramic images were mostly over-
+lapped images. Additionally, the synthesized image is often
+blurred because of the transparency and brightness of the im-
+age change. Therefore, a method was developed to obtain
+clearer images. The implementation of the sharpening pro-
+cess is described in the next section.
+2.5. Creating the Guttae Classifier by U-Net
+The U-Net is a neural network commonly used for image
+segmentation. U-Net uses a convolutional neural network to
+encode an input image as a feature map, subsequently decod-
+ing that feature map to separate specific objects in the input
+image. It is possible to prepare a dataset of corneal images
+and train U-Net using this dataset to generate a model for
+extracting the guttae.
+3. SYSTEM IMPLEMENTATION AND DATA
+APPLICATION
+3.1. Outline
+This study implemented a system to confirm the effec-
+tiveness of the proposed framework.
+Corneal videos ob-
+tained from the FECD mouse model were processed to obtain
+panoramic images of the cornea. Tenengrad was used to ob-
+tain the in-focus images. Finally, two different applications
+were used to generate panoramic images.
+3.2. Mouse Model of FECD
+This study used images of whole corneal endothelial cells
+from the FECD pathology mouse model. A single nucleotide
+mutation in COL8A2 generated these genes, and it has been
+reported that guttae increase over time. The Tissue Engineer-
+ing Laboratory, Graduate School of Biomedical Sciences,
+Doshisha University, provided the images.
+The images
+were taken using a prototype KSSP slit-scanning wide-field
+contact specular microscope (Konan Medical, Inc., Nishi-
+nomiya, Japan), had a resolution of 1620 × 1080 [pixels] at
+a frame rate of 29.9 [fps] in MOV file format. The images
+were taken in the following order: 1) starting from the center
+of the cornea; 2) moving to the top of the cornea; 3) mov-
+ing to the left; 4) filming from the top to the bottom of the
+cornea; 5) moving around the right side; 6) filming from the
+bottom to the center of the cornea. In this study, 94 videos
+were prepared and used.
+3.3. Extraction of in-focus images by Tenengrad
+The focus evaluation index was calculated as follows:
+Tenengrad [12][13] value, which is the gradient of the im-
+age based on the pixel value, is calculated for each region,
+where Gx and Gy are the convolved values of the Sobel op-
+erator of the pixel values in the x-direction and y-direction,
+respectively.
+Φx,y =
+�
+(i,j)∈Ω(x,y)
+(Gx(i, j)2 + Gy(i, j)2)
+(1)
+The Sobel operator is expressed as
+Kx =
+�
+�
+−1
+0
+1
+−2
+0
+2
+−1
+0
+1
+�
+� , Ky =
+�
+�
+−1
+−2
+−1
+0
+0
+0
+1
+2
+1
+�
+�
+The highest value in the quadratic area of a single image
+was considered the focal value for that image. When this
+value was calculated for the entire image, the gradient was
+smaller in the area containing the edge of the corneal en-
+dothelium. In contrast, the gradient increased in the area
+containing corneal endothelial cells. Thus, as the area of
+
+Image integration for 
+panoramic image of the entire cornea
+Learning Phase
+…
+U-Net
+Corneal endothelial
+images including Guttae
+Mask image annotated 
+with Guttae locations
+Dataset 2
+Prediction Phase
+Modeled U-Net
+Video of the entire 
+cornea
+Dataset 1
+• Pre-processing
+• Focused image 
+extraction
+• Feature extraction
+• Image integration
+• Sharpen process of 
+panorama images
+Final panoramic image
+Guttae prediction
+Fig. 1. Overview of the proposed framework
+the rim increases, the Tenengrad value for the entire image
+becomes smaller. This procedure prevents the corneal en-
+dothelium from being excluded from the image even if it is
+appropriately captured.
+3.4. Creating the idealized panoramic artificial CECs
+image data
+To confirm the effectiveness of the panorama synthesis
+software, a set of idealized panoramic artificial corneal en-
+dothelial cell images were created with no blurring or focus
+mismatch on the extracted images. These images were ob-
+tained using the GNU Image Manipulation Program (GIMP).
+These artificial images were created based on the synthesis
+results obtained using the panorama synthesis software de-
+scribed below. A layer was added to the composite image,
+the cell membrane of the corneal endothelium was traced,
+and the areas considered to be guttae were painted black. The
+color of the surrounding endothelial cells was extracted from
+the layer depicting the cell membrane and guttae using the
+color picker function. The layers are filled with the same
+color. This process was applied to the entire cornea to create
+artificial images. For areas where the cell membrane was not
+visible owing to issues such as focus mismatch or blurring,
+the cell membrane was depicted by referring to another im-
+age in which the cell membrane could be observed appropri-
+ately. Idealized panoramic artificial corneal endothelial cell
+images were created that mimicked the distribution and size
+of the guttae, as well as the size and shape of the corneal en-
+dothelial cells. The created image was 1870 × 1080 [pixel]
+in size and was cropped and stored by moving approximately
+10 [pixels] from the center to the top, counterclockwise from
+the top, and counterclockwise from top to bottom, mimick-
+ing the movement of a motion camera.
+3.5. Creating panoramic images
+In the composite process, two types of applications were
+used; the Image Composite Software (ICS) and the panorama
+synthesis algorithm implemented in OpenCV. The algo-
+rithms are explained as follows.
+3.5.1. Image Composite Software (ICS)
+Image Composite Software (K.I. Technology CO., LTD.,
+Yokohama, Kanagawa, Japan) is a panorama compositive
+software created with specifications suitable for image com-
+positing corneal endothelial images. This application was
+used for the composite process and was custom-made. Since
+this is a commercial application, we cannot explain the de-
+tails of its contents due to copyright. The original image is
+not reduced or enlarged when the images are superimposed.
+Additionally, an API to access the original image is provided,
+allowing quick access to the original image of the area of in-
+terest. A flowchart of the panorama compositing process is
+shown in Algorithm 1, where the first image is used as the
+reference image, and the regions that match the first image
+are searched in order of image number. If no match is ob-
+served, the image merged with the reference image is used
+as the reference for the subsequent image, and the process is
+repeated. The image merging is terminated when ten consec-
+utive images are not observed to match the reference image.
+3.5.2. Panorama synthesis algorithm implemented in OpenCV
+Because the details of the process in ICS are not pub-
+licly available, we implemented an algorithm similar to Al-
+gorithm 2 that mimics the ICS process, using OpenCV, an
+open-source computer vision library. The Kth image was the
+closest to the end of the shooting. The jth image and the i+1
+image are matched for SIFT features, and if the two images
+have many similar features, the degree of change between
+the images is calculated and added to the list. The ith image
+is matched to the i + 1 image. If the two images have few
+
+Algorithm 1 Image Composite Software processing details
+1:
+i = 0, j = 0
+2:
+while i + j + 1 ≤ N do
+3:
+A = Images[i]
+4:
+j = 0
+5:
+while j ≤ 10 do
+6:
+B = Images[i+j+1]
+7:
+if Find matching area with A then
+8:
+A = Stitch B onto A
+9:
+i = i + 1, j = 0
+10:
+else
+11:
+j = j + 1
+Algorithm 2 Calculate the difference in coordinates between
+images
+1:
+i = 0
+2:
+coord = [], usedImages = []
+3:
+while j ≤ N do
+4:
+j = i
+5:
+error = 0
+6:
+flag = True
+7:
+while flag do
+8:
+Extract
+SIFT
+features
+of
+Image[j]
+and
+Image[i+1]
+9:
+if Two images could be feature matched then
+10:
+C = cal coord diff(Image[j], Image[i+1])
+11:
+coord.append(C)
+12:
+usedImages.append(Image[j], Image[i+1])
+13:
+i += 1
+14:
+flag = False
+15:
+else
+16:
+error += 1
+17:
+if error ≥ 10 then
+18:
+j += 1
+19:
+else
+20:
+i += 1
+21: usedImages = list(set(usedImages)
+22: return coord, usedImages
+feature points and cannot be matched, add 1 to the values of
+i and f and perform the SIFT feature extraction again. If this
+process fails ten times, add 1 to the value of j and perform
+the process again taking j equivalent to i, that is, j = i. Us-
+ing the above algorithm, the degree of change in coordinates
+between the images used for composition and the images can
+be calculated. The global coordinates of the entire composite
+coordinates can be obtained by setting the smallest values of
+the x and y coordinates to zero and calculating the cumula-
+tive sum till that point.
+3.6. Image sharpening process
+We divided the combined panoramic image into 64 × 64
+[pixel] grid regions. The image number of each composite
+image and the coordinates of the constituent images were
+obtained from the coordinates of the panoramic image. The
+coordinates of the constituent images in the upper-left corner
+of each grid region were obtained. Tenengrad values were
+calculated for the cropped images. The image with the high-
+est Tenengrad value among the cropped images was pasted
+onto a newly created blank image of the same size as the
+panoramic image with the exact extracted coordinates. These
+processes were performed in all regions. The image with the
+highest Tenengrad value among the multiple overlapping im-
+ages was pasted to obtain a clear image.
+3.7. Creating the Guttae Classifier
+3.7.1. Creating Dataset
+A large amount of data is required for training using net-
+works. However, since the amount of data provided in this
+study was small, it was necessary to augment the data. Ad-
+ditionally, due to the large size of the image data, it was nec-
+essary to reduce the size of the images to use them in the
+dataset. Because image resizing results in a loss of infor-
+mation, we developed a new data augmentation method for
+small and large image data. The corneal endothelial cell im-
+age and mask image showing the location of the guttae were
+divided into a grid of 64 × 64 [pixels]. The image was clipped
+three times by 16 [pixels] to the right and three times by 16
+[pixels] to the bottom, thereby shifting the image area. Thus,
+the images were expanded 16 times for a single-grid region.
+Images that did not contain guttae were excluded from the
+dataset.
+3.7.2. Training and Prediction in CNN
+Twenty-one corneal endothelial cell images with a mask
+image showing the location of the guttae were prepared
+(Fig.1 Dataset 2). The Segmentation model PyTorch, which
+is a Python library for implementing segmentation-specific
+CNNs, was used in the experiments. The best encoder back-
+bone is determined using ResNet18, ResNet34, ResNet50,
+VGG11, and VGG16. Twenty-one images were divided into
+two sets of eighteen and three images; eighteen were used
+to develop this model and three were used to determine the
+backbone. The node weights of ImageNet were transferred
+to this model, and fine-tuning was performed. The network
+of U-Net [14] encoders modified to ResNet50 [15] is the best
+backbone. Adam [16] was used as the optimization function
+with an initial learning rate of 1e-4, wherein the loss func-
+tion is the least squares error. Predictions were made on a 64
+× 64 [pixels] image extracted during the sharpening process,
+pasted to the original position, and the location of the guttae
+was predicted for the panoramic image.
+4. RESULTS AND DISCUSSION
+4.1. Comparison of algorithms implemented in ICS and
+OpenCV
+An example of a mouse corneal endothelial cell is shown
+in Fig.2A. The images synthesized using ICS were more cir-
+cular than those synthesized using OpenCV. This result indi-
+cates that the synthesis was performed correctly. On the other
+hand, images synthesized using the OpenCV algorithm were
+not circular and were often pasted in incorrect locations.
+An example of the synthesis result of the algorithm im-
+plemented in OpenCV for artificial cornea data is shown in
+
+(a) Panoramic images by Image Composite Software
+(b) Panoramic images by OpenCV
+A.
+(a) Truth (Artificial cornea image)
+(b) Panoramic image by OpenCV
+B.
+Fig. 2.
+A: Comparison of results with Image Composite Software and software implemented in OpenCV. B: Results with
+OpenCV implemented software on artificial cornea data.(a) shows the composite result and (b) shows the overlap of the
+component images.
+Fig.2B. This is the synthesis result when the focus is per-
+fectly aligned and there are no blurred images. The result is
+almost the same as the correct data, where there is no unnat-
+ural overlap between the composite images, indicating that
+compositing was performed correctly. If the in-focus images
+are well extracted, they can be integrated well using OpenCV.
+Overall, ICS is a more robust method.
+4.2. Synthesis results with ICS
+As previously mentioned, when taking moving images of
+the mouse cornea with the contact specular microscope, the
+images are taken in an upward direction from the center of
+the cornea, then leftward along the edge of the cornea, and
+downward once around the edge. Next, it was photographed
+clockwise and then down to the center. For the 94 videos, the
+left- and right-rotated portions were split, and for each case,
+an integrated image was created using the ICS. The results
+are presented in Table1. An image was classified as ”image
+dropout” if the center of the image was missing, ”distorted
+shape” if the shape was distorted instead of being circular,
+and ”unnatural paste” if the image was pasted in an unnat-
+ural location.
+On the other hand, an image in which the
+shape of the cornea in the combined panoramic image was
+kept circular, the center was not missing, and there were no
+unnaturally pasted parts was classified as a ”success”. The
+number of images in which either the right- or left-rotated
+part of the image was correctly merged was 75. Fig.3 com-
+pares the results of the manual and ICS syntheses. Among
+the videos that failed to be synthesized using ICS, those with
+distorted shapes or unnatural pasting were verified to contain
+frames that deviated from the cornea owing to contamina-
+tion on the specular microscope or a shaking camera. It is
+considered that the stains themselves became feature points,
+and the pasting of the composite part failed. Additionally,
+the position of the image was significantly changed because
+of significant blurring, resulting in an unnatural position for
+Table 1. Synthesis results with ICS
+Result
+Left
+Right
+image dropout
+15
+20
+distorted shape
+6
+12
+unnatural paste
+10
+6
+success
+63
+56
+pasting and distorting the overall shape of the image.
+The results of the ICS and manually composited images
+were almost identical, suggesting that there was no problem
+with the image-compositing algorithm and that the mouse
+cornea was not captured in the first place when the specular
+microscope was used. The image was taken from the cen-
+ter of the mouse cornea in an upward direction, followed by
+leftward rotation along the edge of the cornea, and then a
+downward direction was taken at the point where the image
+had gone around the edge. If there is an area in the center of
+the cornea that has not been photographed at this time, the
+image will be missing.
+4.3. Segmentation of guttae in panoramic images
+The model that detects the guttae location was applied
+to the panoramic images of the videos (Fig.1, Dataset 1).
+Fig.4 shows two prediction examples of the guttae position
+in a panoramic image: (c) and (d) are the prediction results
+in (a) and (b), respectively. The original panoramic images
+were combined using ICS, and the edges of the corneas were
+cropped after sharpening.
+Segmentation was performed on the images synthesized
+from the entire cornea using ICS. Currently, 21 still images
+and a mask image annotated with guttae are used for segmen-
+tation into training and validation datasets. The model was
+evaluated by comparing the validation data with predicted re-
+sults. Fig.4 shows that while the segmentation of the likely
+
+(c) Panoramic image by manually 
+(Panoramic image was integrated Successfully.)
+(d) Panoramic image by manually 
+(There is a hole in the center.)
+(a) Panoramic image by Image Composite Software 
+(Panoramic image was integrated Successfully.)
+(b) Panoramic image by Image Composite Software 
+(There is a hole in the center.)
+Fig. 3.
+Integrated image results by Image Composite Software and manual composite.The left side of each represents the
+composite result, and the right side represents the overlap of the component images.
+(a) Panoramic image with
+corneal rim removed
+(b) Prediction results for 
+guttae location
+(c) Panoramic image with 
+corneal rim removed
+(d) Prediction results for 
+guttae location
+Fig. 4. Two examples of panoramic images and guttae loca-
+tions.
+guttae is successful, it also partially predicts the guttae at the
+edges of the cornea. Further quantitative evaluation is es-
+sential; however, it may pose some limitations for future re-
+search. For this purpose, it is necessary to prepare an image
+to which the Grand Truth of the guttae is assigned.
+4.4. Discussion
+It was observed that the algorithm implemented in
+OpenCV could not correctly synthesize results using mouse
+corneal endothelial cell images. The results showed that the
+images were pasted in unnatural positions compared to those
+obtained using artificial cornea data. The significant differ-
+ence between the mouse corneal endothelial cell image data
+and the artificial cornea data is that, with the artificial cornea
+data, all images are in focus and the images themselves are
+not blurred. For the image data of mouse corneal endothe-
+lial cells, the process of extracting images in focus involved
+extracting images at equal intervals in chronological order,
+which resulted in the extraction of out-of-focus images. In
+contrast, the ICS-based method synthesized the images more
+accurately than the OpenCV-based synthesis software did be-
+cause ICS uses a feature extraction method appropriate for
+corneal endothelial cells.
+By contrast, OpenCV synthesis uses SIFT for feature ex-
+traction. This synthesis is considered to be progressing well.
+Until now, it has not been possible to obtain images of the
+entire cornea because of the narrow imaging range of spec-
+ular microscopy. This study suggests that it is possible to
+obtain a composite image of the entire cornea by extracting
+a relatively focused image from a video of the entire cornea.
+5. CONCLUSIONS AND FUTURE WORK
+The status of the entire cornea is currently inferred from
+the center of diagnosis and observation of corneal endothe-
+lial cells using specular microscopy. If images of the entire
+cornea could be obtained, more studies would be possible.
+In this study, we proposed a framework for generating im-
+ages of the entire cornea from videos captured using contact
+specular microscopy. Focused images were extracted from
+the video and a panoramic composite image was generated.
+Furthermore, we constructed a learning model, U-Net, to ex-
+tract the guttae from the entire image. To study the effec-
+tiveness of the proposed framework, we implemented it and
+applied it to corneal data from a mouse model of FECD. The
+panorama synthesis application used in the implementation
+was our custom-built ICS and the OpenCV algorithm, which
+is an open-source software. Artificial corneal images were
+synthesized with no unnatural aspects in the results. How-
+
+ever, some of the extracted images were not correctly syn-
+thesized if they contained blurred images, and many images
+were correctly synthesized using ICS.
+After the panorama was merged, the image was divided
+into a grid. Majority of the in-focus images were extracted
+and pasted, resulting in a sharper image than the previous
+output obtained using ICS. Using the extracted images within
+the region, we could also predict the guttae location. Al-
+though the implementation and application of the method to
+the data in this study confirmed its effectiveness, few quanti-
+tative evaluations have been performed. Quantitative evalua-
+tion, such as the accuracy of implementation, is an issue for
+the future.
+REFERENCES
+[1]
+Allen O Eghrari, S Amer Riazuddin, and John D
+Gottsch. Fuchs corneal dystrophy. Progress in molecu-
+lar biology and translational science, 134:79–97, 2015.
+[2]
+Gargi Gouranga Nanda and Debasmita Pankaj Alone.
+Current understanding of the pathogenesis of fuchs’ en-
+dothelial corneal dystrophy. Molecular vision, 25:295,
+2019.
+[3]
+Philippe Gain, R´emy Jullienne, Zhiguo He, Mansour
+Aldossary, Sophie Acquart, Fabrice Cognasse, and
+Gilles Thuret. Global survey of corneal transplantation
+and eye banking. JAMA ophthalmology, 134(2):167–
+173, 2016.
+[4]
+Naoki Okumura,
+Shigeru Kinoshita,
+and Noriko
+Koizumi. Application of rho kinase inhibitors for the
+treatment of corneal endothelial diseases. Journal of
+ophthalmology, 2017, 2017.
+[5]
+Naoki Okumura,
+Shigeru Kinoshita,
+and Noriko
+Koizumi. The role of rho kinase inhibitors in corneal
+endothelial dysfunction. Current Pharmaceutical De-
+sign, 23(4):660–666, 2017.
+[6]
+Jay W McLaren, Lori A Bachman, Katrina M Kane,
+and Sanjay V Patel. Objective assessment of the corneal
+endothelium in fuchs’ endothelial dystrophy. Investiga-
+tive ophthalmology & visual science, 55(2):1184–1190,
+2014.
+[7]
+Naoki Okumura, Shohei Yamada, Takeru Nishikawa,
+Kaito Narimoto, Kengo Okamura, Ayaka Izumi, Satoru
+Hiwa, Tomoyuki Hiroyasu, and Noriko Koizumi. U-
+net convolutional neural network for segmenting the
+corneal endothelium in a mouse model of fuchs en-
+dothelial corneal dystrophy. Cornea, 2021.
+[8]
+Takeru Nishikawa, Naoki Okumura, Kaito Narimoto,
+Shohei Yamada, Kengo Okamura, Ayaka Izumi, and
+Noriko Koizumi.
+Deep neural network for the anal-
+ysis of guttae via semi-supervised learning in a fuchs
+endothelial corneal dystrophy mouse model. Investiga-
+tive Ophthalmology & Visual Science, 62(8):826–826,
+2021.
+[9]
+Hiroshi Tanaka, Naoki Okumura, Noriko Koizumi,
+Chie Sotozono, Yasuhiro Sumii, and Shigeru Kinoshita.
+Panoramic view of human corneal endothelial cell layer
+observed by a prototype slit-scanning wide-field con-
+tact specular microscope. British Journal of Ophthal-
+mology, 101(5):655–659, 2017.
+[10] Matthew Brown and David G Lowe.
+Automatic
+panoramic image stitching using invariant features. In-
+ternational journal of computer vision, 74(1):59–73,
+2007.
+[11] David G Lowe. Distinctive image features from scale-
+invariant keypoints. International journal of computer
+vision, 60(2):91–110, 2004.
+[12] Eric Krotkov. Focusing. International Journal of Com-
+puter Vision, 1(3):223–237, 1988.
+[13] Said Pertuz, Domenec Puig, and Miguel Angel Garcia.
+Analysis of focus measure operators for shape-from-
+focus. Pattern Recognition, 46(5):1415–1432, 2013.
+[14] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.
+U-net: Convolutional networks for biomedical image
+segmentation. In International Conference on Medical
+image computing and computer-assisted intervention,
+pages 234–241. Springer, 2015.
+[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian
+Sun. Deep residual learning for image recognition. In
+Proceedings of the IEEE conference on computer vision
+and pattern recognition, pages 770–778, 2016.
+[16] Diederik P Kingma and Jimmy Ba.
+Adam:
+A
+method for stochastic optimization.
+arXiv preprint
+arXiv:1412.6980, 2014.
+

4NE0T4oBgHgl3EQfeQCD/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf,len=346
+page_content='Generating corneal panoramic images from contact specular microscope images  Yusuke Nagira1†, Yuzuha Hara2, Satoru Hiwa2  Naoki Okumura3, Noriko Koizumi3 and Tomoyuki Hiroyasu2  1Graduate School of Life and Medical Sciences, Doshisha University, Japan  2Department of Biomedical Sciences and Informatics, Doshisha University, Japan  3Department of Biomedical Engineering, Faculty of Life and Medical Sciences, Doshisha University, Japan  (Tel: +81-774-65-6020, E-mail: tomo@is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='doshisha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='jp)  Abstract: The contact specular microscope has a wider angle of view than that of the non-contact specular microscope but still  cannot capture an image of the entire cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' To obtain such an image, it is necessary to prepare film on the parts of the image  captured sequentially and combine them to create a complete image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study proposes a framework to automatically generate  an entire corneal image from videos captured using a contact specular microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Relatively focused images were extracted  from the videos and panoramic compositing was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If an entire image can be generated, it is possible to detect guttae  from the image and examine the extent of their presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The system was implemented and the effectiveness of the proposed  framework was examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The system was implemented using custom-made composite software, Image Composite Software  (ICS, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', Japan, internal algorithms not disclosed), and a supervised learning model using U-Net was  used for guttae detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Several images were correctly synthesized when the constructed system was applied to 94 different  corneal videos obtained from Fuchs endothelial corneal dystrophy (FECD) mouse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The implementation and application  of the method to the data in this study confirmed its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Owing to the minimal quantitative evaluation performed, such  as accuracy with implementation, it may pose some limitations for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Keywords: U-Net, Semantic Segmentation, Fuchs Endothelial Corneal Dystrophy, Corneal Endothelial Cell  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' INTRODUCTION  Fuchs endothelial corneal dystrophy (FECD) is a bilat-  eral disease, wherein corneal endothelial cells are unable to  maintain their hexagonal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' It is characterized by the  accelerated loss of corneal endothelial cells with changes in  Descemet’s membrane, resulting in the formation of an ex-  tracellular matrix called guttae [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In the United States, it  is estimated that 4% of people over 40 years of age are af-  fected by the disease, occurring more commonly in women  and more frequently in people in their 40s and 50s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Corneal  endothelial pump function is lost as the disease progresses,  causing corneal edema [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Presently, corneal transplanta-  tion is the only reliable treatment, and FECD accounts for  39% of all corneal transplants performed, making it the most  common cause of corneal transplantation worldwide [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Rho kinase inhibitors have been reported to promote cell  proliferation and adhesion to substrates, inhibit corneal en-  dothelial cell apoptosis, and promote wound healing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' There-  fore, using Rho kinase inhibitor eye drops is a potential  novel treatment approach alternative to corneal transplanta-  tion [4][5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  In drug discovery research for FECD, the state of the  corneal endothelium, such as the guttae, is observed before  and after the drug use and evaluated based on the increase  or decrease in the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Doctors and researchers  widely use specular microscopes to observe the state of the  corneal endothelium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  However, the range of the micro-  scope’s imaging capability is limited and the current practice  estimates the state of the entire cornea from its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  endothelial cell density (ECD) is essential for understanding  † Yusuke Nagira is the presenter of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  the pathogenesis of FECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' However, ECD cannot be mea-  sured accurately owing to the presence of guttae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' hence, the  number of cells is measured manually [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Using a mouse model to demonstrate the pathogenesis of  FECD, studies have been conducted on the segmentation of  guttae using U-Net and the calculation of cell density in areas  excluding guttae [7][8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In previous studies on the panoramic  synthesis of the corneal endothelium, focused images were  extracted manually and stitched together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Panoramic com-  positing of the entire cornea is yet to be performed because  the images were localized panoramic images and did not rep-  resent the entire cornea [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Conventional panorama synthe-  sis software such as AutoStitch [10] does not consider the  order of the input images used for synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Instead, it ex-  tracts the image features using Scale-Invariant Feature Trans-  form (SIFT) [11] and stitches the matching features together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  When pasting an image, it is deformed and scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For im-  ages with similar characteristics, such as corneal endothelial  cells, the position of the image to be pasted may be incor-  rect or the shape of the cells may be deformed owing to the  deformation of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In this case, accurate values can-  not be obtained when calculating the area of the cells or the  percentage of guttae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Therefore, the original image needed  to have as minimal deformations as possible in the combined  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Alternatively, it is necessary to provide a mechanism  to access the original version of the image of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  This study proposes a framework for a system that gen-  erates images of the entire corneal endothelium from videos  obtained using a contact specular microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In the pro-  posed framework, the focused images are extracted from the  video images, feature extraction is performed, and the im-  ages are synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' During synthesis, the system reduces  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='02388v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='IV]  6 Jan 2023  or deforms the original image to the minimum and adds a  feature that allows access to the original image of the area  of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study examined the proposed framework by  building a system using two implementation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fur-  ther, we added a function for automatically detecting guttae  using U-Net, a type of Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study presents a  proposed framework and an example of its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Quantitative evaluation of whole corneal endothelial images  and gut detection is insufficient, which poses a limitation that  must be addressed in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' FRAMEWORK OF CORNEAL PANORAMIC  IMAGE GENERATION FROM CONTACT  SPECULAR MICROSCOPE IMAGES  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Overview of the proposed framework  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1 presents an overview of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  An image of the entire cornea was generated from a video  frame of the cornea captured using a contact-type specu-  lar microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' First, a focused still image was extracted  from the target video image (dataset 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Second, the features  of still images were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Third, the matching feature  points were combined to create a panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Next,  the panoramic image was divided into grid regions and the  most focused image was selected from each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Guttae  were detected in the combined panoramic images using deep  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Additionally, still images of the corneal endothe-  lium containing the guttae and mask images exhibiting the  location of the guttae were used for the model generation  (dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Extraction of in-focus images  A group of still images was extracted from the captured  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The entire corneal endothelium was captured and  converted to a single frame-by-frame image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If there were  N frames in the video, N images were the output in total  that were then divided into five groups in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  The image with the highest in-focus evaluation index was  selected from each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating panoramic images  The algorithm for generating a single panoramic image is  accomplished through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' First, characteris-  tic points in the images were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Subsequently, a curve  connecting the characteristic points was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Here, the  optimal curve connecting the extracted characteristic points  was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This curve was then used to enlarge or interpo-  late the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A panoramic image was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In the fol-  lowing experiments, two algorithms were prepared and the  results were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image sharpening process  The synthesized panoramic images were mostly over-  lapped images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Additionally, the synthesized image is often  blurred because of the transparency and brightness of the im-  age change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Therefore, a method was developed to obtain  clearer images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The implementation of the sharpening pro-  cess is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating the Guttae Classifier by U-Net  The U-Net is a neural network commonly used for image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' U-Net uses a convolutional neural network to  encode an input image as a feature map, subsequently decod-  ing that feature map to separate specific objects in the input  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' It is possible to prepare a dataset of corneal images  and train U-Net using this dataset to generate a model for  extracting the guttae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' SYSTEM IMPLEMENTATION AND DATA  APPLICATION  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Outline  This study implemented a system to confirm the effec-  tiveness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Corneal videos ob-  tained from the FECD mouse model were processed to obtain  panoramic images of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Tenengrad was used to ob-  tain the in-focus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Finally, two different applications  were used to generate panoramic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Mouse Model of FECD  This study used images of whole corneal endothelial cells  from the FECD pathology mouse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A single nucleotide  mutation in COL8A2 generated these genes, and it has been  reported that guttae increase over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The Tissue Engineer-  ing Laboratory, Graduate School of Biomedical Sciences,  Doshisha University, provided the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  The images  were taken using a prototype KSSP slit-scanning wide-field  contact specular microscope (Konan Medical, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', Nishi-  nomiya, Japan), had a resolution of 1620 × 1080 [pixels] at  a frame rate of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='9 [fps] in MOV file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The images  were taken in the following order: 1) starting from the center  of the cornea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 2) moving to the top of the cornea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 3) mov-  ing to the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 4) filming from the top to the bottom of the  cornea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 5) moving around the right side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 6) filming from the  bottom to the center of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In this study, 94 videos  were prepared and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Extraction of in-focus images by Tenengrad  The focus evaluation index was calculated as follows:  Tenengrad [12][13] value, which is the gradient of the im-  age based on the pixel value, is calculated for each region,  where Gx and Gy are the convolved values of the Sobel op-  erator of the pixel values in the x-direction and y-direction,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Φx,y =  �  (i,j)∈Ω(x,y)  (Gx(i, j)2 + Gy(i, j)2)  (1)  The Sobel operator is expressed as  Kx =  �  �  −1  0  1  −2  0  2  −1  0  1  �  � , Ky =  �  �  −1  −2  −1  0  0  0  1  2  1  �  �  The highest value in the quadratic area of a single image  was considered the focal value for that image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' When this  value was calculated for the entire image, the gradient was  smaller in the area containing the edge of the corneal en-  dothelium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In contrast, the gradient increased in the area  containing corneal endothelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Thus, as the area of  Image integration for  panoramic image of the entire cornea  Learning Phase  …  U-Net  Corneal endothelial  images including Guttae  Mask image annotated  with Guttae locations  Dataset 2  Prediction Phase  Modeled U-Net  Video of the entire  cornea  Dataset 1  Pre-processing  Focused image  extraction  Feature extraction  Image integration  Sharpen process of  panorama images  Final panoramic image  Guttae prediction  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Overview of the proposed framework  the rim increases, the Tenengrad value for the entire image  becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This procedure prevents the corneal en-  dothelium from being excluded from the image even if it is  appropriately captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating the idealized panoramic artificial CECs  image data  To confirm the effectiveness of the panorama synthesis  software, a set of idealized panoramic artificial corneal en-  dothelial cell images were created with no blurring or focus  mismatch on the extracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' These images were ob-  tained using the GNU Image Manipulation Program (GIMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  These artificial images were created based on the synthesis  results obtained using the panorama synthesis software de-  scribed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A layer was added to the composite image,  the cell membrane of the corneal endothelium was traced,  and the areas considered to be guttae were painted black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  color of the surrounding endothelial cells was extracted from  the layer depicting the cell membrane and guttae using the  color picker function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The layers are filled with the same  color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This process was applied to the entire cornea to create  artificial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For areas where the cell membrane was not  visible owing to issues such as focus mismatch or blurring,  the cell membrane was depicted by referring to another im-  age in which the cell membrane could be observed appropri-  ately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Idealized panoramic artificial corneal endothelial cell  images were created that mimicked the distribution and size  of the guttae, as well as the size and shape of the corneal en-  dothelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The created image was 1870 × 1080 [pixel]  in size and was cropped and stored by moving approximately  10 [pixels] from the center to the top, counterclockwise from  the top, and counterclockwise from top to bottom, mimick-  ing the movement of a motion camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating panoramic images  In the composite process, two types of applications were  used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' the Image Composite Software (ICS) and the panorama  synthesis algorithm implemented in OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The algo-  rithms are explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image Composite Software (ICS)  Image Composite Software (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Technology CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', LTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=',  Yokohama, Kanagawa, Japan) is a panorama compositive  software created with specifications suitable for image com-  positing corneal endothelial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This application was  used for the composite process and was custom-made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Since  this is a commercial application, we cannot explain the de-  tails of its contents due to copyright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The original image is  not reduced or enlarged when the images are superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Additionally, an API to access the original image is provided,  allowing quick access to the original image of the area of in-  terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A flowchart of the panorama compositing process is  shown in Algorithm 1, where the first image is used as the  reference image, and the regions that match the first image  are searched in order of image number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If no match is ob-  served, the image merged with the reference image is used  as the reference for the subsequent image, and the process is  repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image merging is terminated when ten consec-  utive images are not observed to match the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Panorama synthesis algorithm implemented in OpenCV  Because the details of the process in ICS are not pub-  licly available, we implemented an algorithm similar to Al-  gorithm 2 that mimics the ICS process, using OpenCV, an  open-source computer vision library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The Kth image was the  closest to the end of the shooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The jth image and the i+1  image are matched for SIFT features, and if the two images  have many similar features, the degree of change between  the images is calculated and added to the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The ith image  is matched to the i + 1 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If the two images have few  Algorithm 1 Image Composite Software processing details  1:  i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' j = 0  2:  while i + j + 1 ≤ N do  3:  A = Images[i]  4:  j = 0  5:  while j ≤ 10 do  6:  B = Images[i+j+1]  7:  if Find matching area with A then  8:  A = Stitch B onto A  9:  i = i + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' j = 0  10:  else  11:  j = j + 1  Algorithm 2 Calculate the difference in coordinates between  images  1:  i = 0  2:  coord = [],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' usedImages = []  3:  while j ≤ N do  4:  j = i  5:  error = 0  6:  flag = True  7:  while flag do  8:  Extract  SIFT  features  of  Image[j]  and  Image[i+1]  9:  if Two images could be feature matched then  10:  C = cal coord diff(Image[j],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image[i+1])  11:  coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='append(C)  12:  usedImages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='append(Image[j], Image[i+1])  13:  i += 1  14:  flag = False  15:  else  16:  error += 1  17:  if error ≥ 10 then  18:  j += 1  19:  else  20:  i += 1  21: usedImages = list(set(usedImages)  22: return coord, usedImages  feature points and cannot be matched, add 1 to the values of  i and f and perform the SIFT feature extraction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If this  process fails ten times, add 1 to the value of j and perform  the process again taking j equivalent to i, that is, j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Us-  ing the above algorithm, the degree of change in coordinates  between the images used for composition and the images can  be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The global coordinates of the entire composite  coordinates can be obtained by setting the smallest values of  the x and y coordinates to zero and calculating the cumula-  tive sum till that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image sharpening process  We divided the combined panoramic image into 64 × 64  [pixel] grid regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image number of each composite  image and the coordinates of the constituent images were  obtained from the coordinates of the panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  coordinates of the constituent images in the upper-left corner  of each grid region were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Tenengrad values were  calculated for the cropped images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image with the high-  est Tenengrad value among the cropped images was pasted  onto a newly created blank image of the same size as the  panoramic image with the exact extracted coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' These  processes were performed in all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image with the  highest Tenengrad value among the multiple overlapping im-  ages was pasted to obtain a clear image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating the Guttae Classifier  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating Dataset  A large amount of data is required for training using net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' However, since the amount of data provided in this  study was small, it was necessary to augment the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Ad-  ditionally, due to the large size of the image data, it was nec-  essary to reduce the size of the images to use them in the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Because image resizing results in a loss of infor-  mation, we developed a new data augmentation method for  small and large image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The corneal endothelial cell im-  age and mask image showing the location of the guttae were  divided into a grid of 64 × 64 [pixels].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image was clipped  three times by 16 [pixels] to the right and three times by 16  [pixels] to the bottom, thereby shifting the image area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Thus,  the images were expanded 16 times for a single-grid region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Images that did not contain guttae were excluded from the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Training and Prediction in CNN  Twenty-one corneal endothelial cell images with a mask  image showing the location of the guttae were prepared  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1 Dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The Segmentation model PyTorch, which  is a Python library for implementing segmentation-specific  CNNs, was used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The best encoder back-  bone is determined using ResNet18, ResNet34, ResNet50,  VGG11, and VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Twenty-one images were divided into  two sets of eighteen and three images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' eighteen were used  to develop this model and three were used to determine the  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The node weights of ImageNet were transferred  to this model, and fine-tuning was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The network  of U-Net [14] encoders modified to ResNet50 [15] is the best  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Adam [16] was used as the optimization function  with an initial learning rate of 1e-4, wherein the loss func-  tion is the least squares error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Predictions were made on a 64  × 64 [pixels] image extracted during the sharpening process,  pasted to the original position, and the location of the guttae  was predicted for the panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Comparison of algorithms implemented in ICS and  OpenCV  An example of a mouse corneal endothelial cell is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The images synthesized using ICS were more cir-  cular than those synthesized using OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This result indi-  cates that the synthesis was performed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' On the other  hand, images synthesized using the OpenCV algorithm were  not circular and were often pasted in incorrect locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  An example of the synthesis result of the algorithm im-  plemented in OpenCV for artificial cornea data is shown in  (a) Panoramic images by Image Composite Software  (b) Panoramic images by OpenCV  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  (a) Truth (Artificial cornea image)  (b) Panoramic image by OpenCV  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  A: Comparison of results with Image Composite Software and software implemented in OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' B: Results with  OpenCV implemented software on artificial cornea data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' (a) shows the composite result and (b) shows the overlap of the  component images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This is the synthesis result when the focus is per-  fectly aligned and there are no blurred images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The result is  almost the same as the correct data, where there is no unnat-  ural overlap between the composite images, indicating that  compositing was performed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If the in-focus images  are well extracted, they can be integrated well using OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Overall, ICS is a more robust method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Synthesis results with ICS  As previously mentioned, when taking moving images of  the mouse cornea with the contact specular microscope, the  images are taken in an upward direction from the center of  the cornea, then leftward along the edge of the cornea, and  downward once around the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Next, it was photographed  clockwise and then down to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For the 94 videos, the  left- and right-rotated portions were split, and for each case,  an integrated image was created using the ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The results  are presented in Table1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' An image was classified as ”image  dropout” if the center of the image was missing, ”distorted  shape” if the shape was distorted instead of being circular,  and ”unnatural paste” if the image was pasted in an unnat-  ural location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  On the other hand, an image in which the  shape of the cornea in the combined panoramic image was  kept circular, the center was not missing, and there were no  unnaturally pasted parts was classified as a ”success”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  number of images in which either the right- or left-rotated  part of the image was correctly merged was 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3 com-  pares the results of the manual and ICS syntheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Among  the videos that failed to be synthesized using ICS, those with  distorted shapes or unnatural pasting were verified to contain  frames that deviated from the cornea owing to contamina-  tion on the specular microscope or a shaking camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' It is  considered that the stains themselves became feature points,  and the pasting of the composite part failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Additionally,  the position of the image was significantly changed because  of significant blurring, resulting in an unnatural position for  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Synthesis results with ICS  Result  Left  Right  image dropout  15  20  distorted shape  6  12  unnatural paste  10  6  success  63  56  pasting and distorting the overall shape of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  The results of the ICS and manually composited images  were almost identical, suggesting that there was no problem  with the image-compositing algorithm and that the mouse  cornea was not captured in the first place when the specular  microscope was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image was taken from the cen-  ter of the mouse cornea in an upward direction, followed by  leftward rotation along the edge of the cornea, and then a  downward direction was taken at the point where the image  had gone around the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If there is an area in the center of  the cornea that has not been photographed at this time, the  image will be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Segmentation of guttae in panoramic images  The model that detects the guttae location was applied  to the panoramic images of the videos (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1, Dataset 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4 shows two prediction examples of the guttae position  in a panoramic image: (c) and (d) are the prediction results  in (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The original panoramic images  were combined using ICS, and the edges of the corneas were  cropped after sharpening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Segmentation was performed on the images synthesized  from the entire cornea using ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Currently, 21 still images  and a mask image annotated with guttae are used for segmen-  tation into training and validation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The model was  evaluated by comparing the validation data with predicted re-  sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4 shows that while the segmentation of the likely  (c) Panoramic image by manually  (Panoramic image was integrated Successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  (d) Panoramic image by manually  (There is a hole in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  (a) Panoramic image by Image Composite Software  (Panoramic image was integrated Successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  (b) Panoramic image by Image Composite Software  (There is a hole in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Integrated image results by Image Composite Software and manual composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='The left side of each represents the  composite result, and the right side represents the overlap of the component images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  (a) Panoramic image with  corneal rim removed  (b) Prediction results for  guttae location  (c) Panoramic image with  corneal rim removed  (d) Prediction results for  guttae location  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Two examples of panoramic images and guttae loca-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  guttae is successful, it also partially predicts the guttae at the  edges of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Further quantitative evaluation is es-  sential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' however, it may pose some limitations for future re-  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For this purpose, it is necessary to prepare an image  to which the Grand Truth of the guttae is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Discussion  It was observed that the algorithm implemented in  OpenCV could not correctly synthesize results using mouse  corneal endothelial cell images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The results showed that the  images were pasted in unnatural positions compared to those  obtained using artificial cornea data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The significant differ-  ence between the mouse corneal endothelial cell image data  and the artificial cornea data is that, with the artificial cornea  data, all images are in focus and the images themselves are  not blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For the image data of mouse corneal endothe-  lial cells, the process of extracting images in focus involved  extracting images at equal intervals in chronological order,  which resulted in the extraction of out-of-focus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In  contrast, the ICS-based method synthesized the images more  accurately than the OpenCV-based synthesis software did be-  cause ICS uses a feature extraction method appropriate for  corneal endothelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  By contrast, OpenCV synthesis uses SIFT for feature ex-  traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This synthesis is considered to be progressing well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Until now, it has not been possible to obtain images of the  entire cornea because of the narrow imaging range of spec-  ular microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study suggests that it is possible to  obtain a composite image of the entire cornea by extracting  a relatively focused image from a video of the entire cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' CONCLUSIONS AND FUTURE WORK  The status of the entire cornea is currently inferred from  the center of diagnosis and observation of corneal endothe-  lial cells using specular microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If images of the entire  cornea could be obtained, more studies would be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  In this study, we proposed a framework for generating im-  ages of the entire cornea from videos captured using contact  specular microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Focused images were extracted from  the video and a panoramic composite image was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Furthermore, we constructed a learning model, U-Net, to ex-  tract the guttae from the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' To study the effec-  tiveness of the proposed framework, we implemented it and  applied it to corneal data from a mouse model of FECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  panorama synthesis application used in the implementation  was our custom-built ICS and the OpenCV algorithm, which  is an open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Artificial corneal images were  synthesized with no unnatural aspects in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' How-  ever, some of the extracted images were not correctly syn-  thesized if they contained blurred images, and many images  were correctly synthesized using ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  After the panorama was merged, the image was divided  into a grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Majority of the in-focus images were extracted  and pasted, resulting in a sharper image than the previous  output obtained using ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Using the extracted images within  the region, we could also predict the guttae location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Al-  though the implementation and application of the method to  the data in this study confirmed its effectiveness, few quanti-  tative evaluations have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Quantitative evalua-  tion, such as the accuracy of implementation, is an issue for  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  REFERENCES  [1]  Allen O Eghrari, S Amer Riazuddin, and John D  Gottsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fuchs corneal dystrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
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+page_content='  [2]  Gargi Gouranga Nanda and Debasmita Pankaj Alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Current understanding of the pathogenesis of fuchs’ en-  dothelial corneal dystrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Molecular vision, 25:295,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [3]  Philippe Gain, R´emy Jullienne, Zhiguo He, Mansour  Aldossary, Sophie Acquart, Fabrice Cognasse, and  Gilles Thuret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
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+page_content='  [4]  Naoki Okumura,  Shigeru Kinoshita,  and Noriko  Koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Application of rho kinase inhibitors for the  treatment of corneal endothelial diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Journal of  ophthalmology, 2017, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [5]  Naoki Okumura,  Shigeru Kinoshita,  and Noriko  Koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The role of rho kinase inhibitors in corneal  endothelial dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Current Pharmaceutical De-  sign, 23(4):660–666, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
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+arXiv:2301.02390v1  [eess.IV]  6 Jan 2023
+Deep-learning models in medical image analysis:
+Detection of esophagitis from the Kvasir Dataset
+Kyoka Yoshioka1†, Kensuke Tanioka2, Satoru Hiwa2 and Tomoyuki Hiroyasu2
+1Graduate School of Life and Medical Sciences, Doshisha University, Kyoto, Japan
+2Department of Biomedical Sciences and Informatics, Doshisha University, Kyoto, Japan
+(Tel: +81-774-65-6020; E-mail: [email protected])
+Abstract: Early detection of esophagitis is important because this condition can progress to cancer if left untreated. However,
+the accuracies of different deep learning models in detecting esophagitis have yet to be compared. Thus, this study aimed to
+compare the accuracies of convolutional neural network models (GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3) in
+detecting esophagitis from the open Kvasir dataset of endoscopic images. Results showed that among the models, GoogLeNet
+achieved the highest F1-scores. Based on the average of true positive rate, MobileNet V3 predicted esophagitis more confidently
+than the other models. The results obtained using the models were also compared with those obtained using SHapley Additive
+exPlanations and Gradient-weighted Class Activation Mapping.
+Keywords: Kvasir dataset, Deep Learning, Convolutional Neural Networks, Gradient-Weighted Class Activation Mapping,
+SHAP, SHapley Additive exPlanation
+1. INTRODUCTION
+With the development of artificial intelligence (AI), sev-
+eral studies have focused on the application of this technol-
+ogy in the medical field.
+In gastroenterology, AI is used
+to detect inflammation, polyps, and stomach cancer and de-
+velop systems that can automatically determine the severity
+of symptoms [1] [2] [3] [4]. AI models are expected to im-
+prove diagnostic accuracy and reduce medical costs by pre-
+venting misdiagnosis by humans.
+Various deep learning and AI models, including deep
+learning convolutional neural network (CNN) models, have
+been proposed and used for medical image recognition and
+analysis. However, these models differ in accuracy, and com-
+paring this aspect is important to identify which model is
+suitable for a specific application in endoscopic imaging.
+The z-line is an anatomic landmark located posterior to
+the stomach and esophagus. Esophagitis is an inflammation
+of the esophagus that appears as a break in the esophageal
+mucosa relative to the z-line [5]. The z-line and esophagitis
+can be described as normal and diseased conditions, respec-
+tively. Early detection of esophagitis is necessary because
+this condition can cause complications (e.g., esophageal ul-
+cer, bleeding, and stricture) and progress to cancer if left
+untreated. Therefore, distinguishing between the z-line and
+esophagitis is necessary. However, this procedure is difficult
+[6]. In addition, the accuracies of various models in detecting
+esophagitis have yet to be compared.
+Thus, this study aimed to compare the accuracies of sev-
+eral CNN models, including GoogLeNet [7], ResNet-50 [8],
+MobileNet V2 [9], and MobileNet V3 [10], in identifying
+z-lines and esophagitis in endoscopic images from the open
+Kvasir dataset. These models have received considerable at-
+tention in recent years after winning in the ImageNet Large
+Scale Visual Recognition Challenge (ILSVRC), a competi-
+tion using a large image recognition dataset. The results ob-
+† Kyoka Yoshioka is the presenter of this paper.
+tained by the four CNN models were compared. The training
+models were also compared with the explainable artificial in-
+telligence (XAI) methods Gradient-weighted Class Activa-
+tion Mapping (Grad-CAM) [11] and SHapley Additive ex-
+Planations (SHAP) [12].
+2. DEEP LEARNING IN MEDICAL IMAGE
+ANALYSIS
+2.1. Typical architecture for image classification
+CNN is a deep learning method specialized for image
+recognition. It is widely used for identifying lesion sites in
+medical images. It combines a convolutional layer with a
+pooling layer and finally iterates through all the combined
+layers to generate the results. In this study, we compared
+the results of different CNN models used for site identifi-
+cation in medical images. The CNN models used included
+GoogleNet and ResNet, the successive winning models of
+ILSVRC, and MobileNet V2 and MobileNet V3, which have
+attracted considerable attention in recent years because of
+their small computational and memory.
+2.1.1. GoogLeNet
+GoogLeNet was the winning model at ILSVRC in 2014
+The model consists of an Inception module, 1×1 convolu-
+tion, auxiliary loss, and global average pooling. GoogLeNet
+can be multi-layered using the Inception module, but 1×1
+convolution is performed before each convolution calcula-
+tion to reduce dimensionality resulting from the large num-
+ber of parameters. The Inception module helps process data
+using multiple filters in parallel. The fully connected layer
+is removed to increase the width and depth of the network,
+average pooling is used instead of the fully connected layer
+to avoid gradient loss, and class classification is performed
+on sub-networks branched from the middle of the network
+by auxiliary loss [7].
+
+2.1.2. ResNet
+ResNet was the winning model at the ILSVRC in 2015.
+The problem of learning not progressing due to gradient loss
+and degradation problems was solved using a method called
+Residual Block, which uses 152 very deep layers to solve the
+problem. The key features of this model are residual block
+and batch normalization using shortcut connection. ResNet
+has several models with different layer depths. ResNet-50
+shows higher accuracy than GoogLeNet in ImageNet clas-
+sification [8]. However, ResNet-50 requires about twice as
+many parameters as GoogLeNet.
+2.1.3. MobileNet V2
+MobileNet is a small computationally and memory model
+that can adjust the trade-off between accuracy and compu-
+tational load. Depthwise separable convolution decomposes
+the convolution layer into depthwise and pointwise convolu-
+tion for computation. This mechanism reduces the compu-
+tation cost. Furthermore, V2 introduces expand/projection
+layers and inverted residual blocks. Expand/projection lay-
+ers rapidly increase or decrease the number of channels. Mo-
+bileNet V2 achieves comparable accuracy to GoogLeNet and
+ResNet-50 in ImageNet classification while significantly re-
+ducing the number of parameters [9].
+2.1.4. MobileNet V3
+MobileNet V3 is an improved version of MobileNet V2,
+introducing a squeeze-and-excite structure (SE-block) in the
+inverted residual block, one of the features of MobileNet
+V2. SE-block improves the expressiveness of the model by
+weighting information in the channel direction [13]. Com-
+pared with V2, MobileNet V3 shows more accurate Im-
+ageNet classification while shortening total inference time
+[10].
+2.2. Explainable AI (XAI)
+The CNN models were compared with XAI methods
+Grad-CAM and SHAP. The Discussion section explains the
+results obtained using these techniques.
+2.2.1. Grad CAM
+Grad-CAM displays a color map of the area the CNN is
+gazing at for classification [11]. It is based on the fact that
+variables with large gradients in the output values of the pre-
+dicted class are essential for classification prediction. The
+gradient of each input image pixel with respect to the output
+value of the prediction class in the last convolution layer is
+used.
+2.2.2. SHAP
+SHAP calculates, for each predicted value, how each char-
+acteristic variable affects that prediction [12]. This analysis
+allows us to visualize the impact of an increase or decrease
+in the value of a given characteristic variable.
+3. MATERIALS AND METHODS
+CNN models GoogLeNet, ResNet-50, MobileNet V2, and
+MobileNet V3 were employed to detect esophagitis from the
+open Kvasir dataset of endoscopic images, and their results
+were compared.
+3.1. Kvasir dataset
+The Kvasir dataset is a collection of endoscopic images of
+the gastrointestinal tract. It was annotated and validated by
+certified endoscopists. The dataset was made available in the
+fall of 2017 through the Medical Multimedia Challenge pro-
+vided by MediaEval. It includes anatomical landmarks (py-
+lorus, z-line, and cecum), disease states (esophagitis, ulcera-
+tive colitis, and polyps), and medical procedures (dyed lifted
+polyps and dyed resection margins). The resolution of the
+images from the Kvasir dataset with these eight classes varies
+from 720×576 pixels to 1920×1072 pixels. Each image has
+a different shooting angle, resolution, brightness, magnifica-
+tion, and center point.
+3.2. Prepossessing
+Image prepossessing was performed before training the
+models. Edge artifacts and annotations that interfere with
+learning during the analysis of medical images were re-
+moved. A mask image was created, where pixels with lu-
+minance values below a certain threshold were set to 0. The
+opening process was applied to the mask image to remove the
+annotations. The image was cropped using this final mask
+image to obtain the target area. This process was performed
+on all data.
+Each image in the dataset has a different resolution. All
+images were resized to 224×224 pixels by bilinear comple-
+tion and optimized for deep learning input. In addition to
+these processes, data augmentation was performed on the
+data used for learning. We applied two types of data aug-
+mentation: horizontal and vertical flip.
+3.3. Cross Validation
+A total of 1000 image data sets containing z-lines and
+esophagitis were partitioned into test, training, and validation
+data. First, 25% (n = 250) of the total data were randomly se-
+lected to generate test data. Of the remaining data (75%, n =
+750), 50% (n = 500) was used for training and 25% (n = 250)
+for validation.
+The inner loop consisted of training and validation data.
+The model was trained using the training data, and parame-
+ters such as the optimal number of epochs were determined
+using the validation data. Thus, four training models were
+generated. The test data of each model were evaluated, and
+the average of discrimination accuracy of the four times was
+used as the evaluation value of the CNN model. The test,
+training, and validation data were each partitioned to main-
+tain the class proportions.
+3.4. CNN models
+PyTorch was used for the implementation of GoogLeNet,
+ResNet-50, MobileNet V2, and MobileNet V3.
+The ini-
+tial values of all model parameters were pre-trained by Ima-
+geNet, and the models were trained by fine tuning.
+For all models, the Adam optimizer was used for training.
+The batch size was five, and the maximum number of epochs
+
+was 100. The cross-entropy error shown in equation (1) was
+used as the loss function.
+E(x)
+=
+−
+N
+�
+n=1
+K
+�
+k=1
+dnk log yk(xn; w)
+(1)
+3.5. Evaluation Function
+Five evaluation indices were used in this experiment: ac-
+curacy, precision, recall, specificity, and F1-score. These
+metrics were calculated using the confusion matrix shown
+in Table 1.
+Table 1. Confusion matrix for a two-class problem
+Predicted Class
+(Positive Class)
+Predicted Class
+(Negative Class)
+Actual Class
+(Positive Class)
+True Positive
+False Negative
+Actual Class
+(Negative Class)
+False Positive
+True Negative
+In this experiment, the z-line and esophagitis were judged
+as the negative and positive classes, respectively. In other
+words, data judged to be esophagitis and z-line by the learn-
+ing model were designated true positive (TP) and false neg-
+ative (FN), respectively.
+Meanwhile, data determined to
+be esophagitis and z-line by the training model were des-
+ignated false positive (FP) and true negative (TN), respec-
+tively. Based on the values of TP, FP, TN, and FN obtained
+from the confusion matrix, the accuracy, precision, recall,
+specificity, and F1-score of the models were calculated using
+Equations(2) to (6).
+Accuracy =
+T P + T N
+T P + FP + FN + T N
+(2)
+Precision =
+T P
+T P + FP
+(3)
+Recall =
+T P
+T P + FN
+(4)
+Specificity =
+T N
+T N + FP
+(5)
+F1 score =
+2T N
+2T P + FP + FN
+(6)
+4. RESULTS AND DISCUSSIONS
+4.1. Performance comparison between different archi-
+tecture
+The evaluation indices obtained from the experiments are
+shown in Table 2.
+The F1-score results in Table 2 show that GoogLeNet was
+the best among the four models. In other words, GoogLeNet
+was more reliable in predicting esophagitis than the other
+models. Meanwhile, MobileNet V3 showed the highest pre-
+cision and specificity. In other words, MobileNet V3 was
+the most accurate among the tested models for z-line predic-
+tion. From a medical point of view, an ideal model should be
+Table 2. Performance comparison between different
+architecture
+Model
+ACC
+PREC
+REC
+SPEC
+F1
+GoogLeNet
+0.846
+0.859
+0.830
+0.862
+0.843
+MobileNet V3
+0.842
+0.901
+0.776
+0.908
+0.831
+ResNet-50
+0.833
+0.865
+0.792
+0.874
+0.826
+MobileNet V2
+0.830
+0.852
+0.800
+0.860
+0.825
+likely to distinguish esophagitis with severe symptoms from
+the z-line.
+The average of TP rate were 0.950, 0.923, 0.892, and
+0.841 for MobileNet V3, MobileNet V2, GoogLeNet, and
+ResNet-50, respectively. MobileNet V3 predicted esophagi-
+tis with more confidence than the other models.
+4.2. GoogLeNet analysis
+Grad-CAM and SHAP were applied to the learned model,
+and what kind of the model was created was discussed.
+Fig.1 shows an example of the image results in the case of
+TP predicted by GoogLeNet. In the Grad-CAM results, red
+indicates the most potent activation, and blue indicates the
+weakest activation. In the SHAP results, the SHAP values
+of the patches were computed and rendered in a color map:
+a positive SHHAP value (red) indicates that the class is sup-
+ported. By contrast, a negative SHAP value (blue) indicates
+that the class is rejected.
+Tearing the esophageal mucosa against the z-line is a
+feature of esophagitis.
+According to Fig.1, the results of
+Grad-CAM and SHAP showed that the learned model of
+GoogLeNet can makes predictions focusing on the clinically
+significant aspects of esophagitis images. The GoogLeNet
+model learned the findings that are important for diagnosing
+esophagitis. Comparison results showed that SHAP captured
+the location of multiple mucosal tears in the image more ac-
+curately than Grad-CAM.
+Fig.2 shows the results of applying Grad-CAM and SHAP
+in the FN case. The following can be observed from the re-
+sults of Grad-CAM and SHAP for Fig.2, respectively. In the
+Grad-CAM results, most areas in the image are shown as
+activated regions. Areas that provide the basis for the pre-
+diction are difficult to identify because of the gradient satu-
+ration in the Grad-CAM calculation. In the SHAP results,
+the inflammatory areas of the input image are indicated by
+blue pixels. Blue pixels indicate features that have a negative
+contribution to the prediction. In other words, although the
+model incorrectly identified esophagitis as a z-line, the model
+recognized that areas in the image negatively contributed to
+the z-line decision.
+4.3. MobileNet V3 analysis
+One hundred images were determined to be TP in the
+MobileNet V3 model.
+The SHAP results for the images
+judged to have the highest and lowest probabilities of being
+esophagitis are shown in Fig.3.
+As shown in Fig.3, in cases with a high prediction proba-
+bility, some features may have a negative contribution to the
+
+(a) Raw image
+(b) Grad-CAM
+(c) SHAP
+Fig. 1. True Positive Pattern
+(a) Raw image
+(b) Grad-CAM
+(c) SHAP
+Fig. 2. False Negative Pattern
+Fig. 3. First image predicted positive with 1.000 probability, and second image predicted positive with 0.524 probability.
+prediction. Many features showing negative contributions
+can be identified in the images with low prediction proba-
+bility for Fig.3. In this case, the prediction probability may
+be low.
+5. CONCLUSIONS
+We compared the accuracies of CNN models, including
+GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3,
+in identifying z-line and esophagitis in endoscopic images
+from the open Kvasir dataset.
+Among the four models,
+GoogLeNet had the highest F1-score, and MobileNet V3
+had the highest average TP rate. These results suggest that
+GoogLeNet performs better than state-of-the-art CNN mod-
+els in medical image recognition. In addition, MoblieNet V3
+is a cost-effective model because of its low memory and short
+training time. Each model was analyzed and compared with
+Grad-CAM, and SHAP. Other models, datasets, and model
+analyses are warranted for verification.
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+

4tE0T4oBgHgl3EQfegAX/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='02390v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='IV]  6 Jan 2023  Deep-learning models in medical image analysis:  Detection of esophagitis from the Kvasir Dataset  Kyoka Yoshioka1†, Kensuke Tanioka2, Satoru Hiwa2 and Tomoyuki Hiroyasu2  1Graduate School of Life and Medical Sciences, Doshisha University, Kyoto, Japan  2Department of Biomedical Sciences and Informatics, Doshisha University, Kyoto, Japan  (Tel: +81-774-65-6020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' E-mail: tomo@is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='doshisha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='jp)  Abstract: Early detection of esophagitis is important because this condition can progress to cancer if left untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However,  the accuracies of different deep learning models in detecting esophagitis have yet to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Thus, this study aimed to  compare the accuracies of convolutional neural network models (GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3) in  detecting esophagitis from the open Kvasir dataset of endoscopic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Results showed that among the models, GoogLeNet  achieved the highest F1-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Based on the average of true positive rate, MobileNet V3 predicted esophagitis more confidently  than the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The results obtained using the models were also compared with those obtained using SHapley Additive  exPlanations and Gradient-weighted Class Activation Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Keywords: Kvasir dataset, Deep Learning, Convolutional Neural Networks, Gradient-Weighted Class Activation Mapping,  SHAP, SHapley Additive exPlanation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' INTRODUCTION  With the development of artificial intelligence (AI), sev-  eral studies have focused on the application of this technol-  ogy in the medical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  In gastroenterology, AI is used  to detect inflammation, polyps, and stomach cancer and de-  velop systems that can automatically determine the severity  of symptoms [1] [2] [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' AI models are expected to im-  prove diagnostic accuracy and reduce medical costs by pre-  venting misdiagnosis by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Various deep learning and AI models, including deep  learning convolutional neural network (CNN) models, have  been proposed and used for medical image recognition and  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However, these models differ in accuracy, and com-  paring this aspect is important to identify which model is  suitable for a specific application in endoscopic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The z-line is an anatomic landmark located posterior to  the stomach and esophagus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Esophagitis is an inflammation  of the esophagus that appears as a break in the esophageal  mucosa relative to the z-line [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The z-line and esophagitis  can be described as normal and diseased conditions, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Early detection of esophagitis is necessary because  this condition can cause complications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=', esophageal ul-  cer, bleeding, and stricture) and progress to cancer if left  untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Therefore, distinguishing between the z-line and  esophagitis is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However, this procedure is difficult  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In addition, the accuracies of various models in detecting  esophagitis have yet to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Thus, this study aimed to compare the accuracies of sev-  eral CNN models, including GoogLeNet [7], ResNet-50 [8],  MobileNet V2 [9], and MobileNet V3 [10], in identifying  z-lines and esophagitis in endoscopic images from the open  Kvasir dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' These models have received considerable at-  tention in recent years after winning in the ImageNet Large  Scale Visual Recognition Challenge (ILSVRC), a competi-  tion using a large image recognition dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The results ob-  † Kyoka Yoshioka is the presenter of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  tained by the four CNN models were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The training  models were also compared with the explainable artificial in-  telligence (XAI) methods Gradient-weighted Class Activa-  tion Mapping (Grad-CAM) [11] and SHapley Additive ex-  Planations (SHAP) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' DEEP LEARNING IN MEDICAL IMAGE  ANALYSIS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Typical architecture for image classification  CNN is a deep learning method specialized for image  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It is widely used for identifying lesion sites in  medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It combines a convolutional layer with a  pooling layer and finally iterates through all the combined  layers to generate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In this study, we compared  the results of different CNN models used for site identifi-  cation in medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The CNN models used included  GoogleNet and ResNet, the successive winning models of  ILSVRC, and MobileNet V2 and MobileNet V3, which have  attracted considerable attention in recent years because of  their small computational and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' GoogLeNet  GoogLeNet was the winning model at ILSVRC in 2014  The model consists of an Inception module, 1×1 convolu-  tion, auxiliary loss, and global average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' GoogLeNet  can be multi-layered using the Inception module, but 1×1  convolution is performed before each convolution calcula-  tion to reduce dimensionality resulting from the large num-  ber of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The Inception module helps process data  using multiple filters in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The fully connected layer  is removed to increase the width and depth of the network,  average pooling is used instead of the fully connected layer  to avoid gradient loss, and class classification is performed  on sub-networks branched from the middle of the network  by auxiliary loss [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' ResNet  ResNet was the winning model at the ILSVRC in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The problem of learning not progressing due to gradient loss  and degradation problems was solved using a method called  Residual Block, which uses 152 very deep layers to solve the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The key features of this model are residual block  and batch normalization using shortcut connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' ResNet  has several models with different layer depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' ResNet-50  shows higher accuracy than GoogLeNet in ImageNet clas-  sification [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However, ResNet-50 requires about twice as  many parameters as GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V2  MobileNet is a small computationally and memory model  that can adjust the trade-off between accuracy and compu-  tational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Depthwise separable convolution decomposes  the convolution layer into depthwise and pointwise convolu-  tion for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' This mechanism reduces the compu-  tation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Furthermore, V2 introduces expand/projection  layers and inverted residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Expand/projection lay-  ers rapidly increase or decrease the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Mo-  bileNet V2 achieves comparable accuracy to GoogLeNet and  ResNet-50 in ImageNet classification while significantly re-  ducing the number of parameters [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V3  MobileNet V3 is an improved version of MobileNet V2,  introducing a squeeze-and-excite structure (SE-block) in the  inverted residual block, one of the features of MobileNet  V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' SE-block improves the expressiveness of the model by  weighting information in the channel direction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Com-  pared with V2, MobileNet V3 shows more accurate Im-  ageNet classification while shortening total inference time  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Explainable AI (XAI)  The CNN models were compared with XAI methods  Grad-CAM and SHAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The Discussion section explains the  results obtained using these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Grad CAM  Grad-CAM displays a color map of the area the CNN is  gazing at for classification [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It is based on the fact that  variables with large gradients in the output values of the pre-  dicted class are essential for classification prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The  gradient of each input image pixel with respect to the output  value of the prediction class in the last convolution layer is  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' SHAP  SHAP calculates, for each predicted value, how each char-  acteristic variable affects that prediction [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' This analysis  allows us to visualize the impact of an increase or decrease  in the value of a given characteristic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MATERIALS AND METHODS  CNN models GoogLeNet, ResNet-50, MobileNet V2, and  MobileNet V3 were employed to detect esophagitis from the  open Kvasir dataset of endoscopic images, and their results  were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Kvasir dataset  The Kvasir dataset is a collection of endoscopic images of  the gastrointestinal tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It was annotated and validated by  certified endoscopists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The dataset was made available in the  fall of 2017 through the Medical Multimedia Challenge pro-  vided by MediaEval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It includes anatomical landmarks (py-  lorus, z-line, and cecum), disease states (esophagitis, ulcera-  tive colitis, and polyps), and medical procedures (dyed lifted  polyps and dyed resection margins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The resolution of the  images from the Kvasir dataset with these eight classes varies  from 720×576 pixels to 1920×1072 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Each image has  a different shooting angle, resolution, brightness, magnifica-  tion, and center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Prepossessing  Image prepossessing was performed before training the  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Edge artifacts and annotations that interfere with  learning during the analysis of medical images were re-  moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' A mask image was created, where pixels with lu-  minance values below a certain threshold were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The  opening process was applied to the mask image to remove the  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The image was cropped using this final mask  image to obtain the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' This process was performed  on all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Each image in the dataset has a different resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' All  images were resized to 224×224 pixels by bilinear comple-  tion and optimized for deep learning input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In addition to  these processes, data augmentation was performed on the  data used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' We applied two types of data aug-  mentation: horizontal and vertical flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Cross Validation  A total of 1000 image data sets containing z-lines and  esophagitis were partitioned into test, training, and validation  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' First, 25% (n = 250) of the total data were randomly se-  lected to generate test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Of the remaining data (75%, n =  750), 50% (n = 500) was used for training and 25% (n = 250)  for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The inner loop consisted of training and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The model was trained using the training data, and parame-  ters such as the optimal number of epochs were determined  using the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Thus, four training models were  generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The test data of each model were evaluated, and  the average of discrimination accuracy of the four times was  used as the evaluation value of the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The test,  training, and validation data were each partitioned to main-  tain the class proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' CNN models  PyTorch was used for the implementation of GoogLeNet,  ResNet-50, MobileNet V2, and MobileNet V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The ini-  tial values of all model parameters were pre-trained by Ima-  geNet, and the models were trained by fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  For all models, the Adam optimizer was used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The batch size was five, and the maximum number of epochs  was 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The cross-entropy error shown in equation (1) was  used as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  E(x)  =  −  N  �  n=1  K  �  k=1  dnk log yk(xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' w)  (1)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Evaluation Function  Five evaluation indices were used in this experiment: ac-  curacy, precision, recall, specificity, and F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' These  metrics were calculated using the confusion matrix shown  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Confusion matrix for a two-class problem  Predicted Class  (Positive Class)  Predicted Class  (Negative Class)  Actual Class  (Positive Class)  True Positive  False Negative  Actual Class  (Negative Class)  False Positive  True Negative  In this experiment, the z-line and esophagitis were judged  as the negative and positive classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other  words, data judged to be esophagitis and z-line by the learn-  ing model were designated true positive (TP) and false neg-  ative (FN), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Meanwhile, data determined to  be esophagitis and z-line by the training model were des-  ignated false positive (FP) and true negative (TN), respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Based on the values of TP, FP, TN, and FN obtained  from the confusion matrix, the accuracy, precision, recall,  specificity, and F1-score of the models were calculated using  Equations(2) to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Accuracy =  T P + T N  T P + FP + FN + T N  (2)  Precision =  T P  T P + FP  (3)  Recall =  T P  T P + FN  (4)  Specificity =  T N  T N + FP  (5)  F1 score =  2T N  2T P + FP + FN  (6)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' RESULTS AND DISCUSSIONS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Performance comparison between different archi-  tecture  The evaluation indices obtained from the experiments are  shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The F1-score results in Table 2 show that GoogLeNet was  the best among the four models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other words, GoogLeNet  was more reliable in predicting esophagitis than the other  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Meanwhile, MobileNet V3 showed the highest pre-  cision and specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other words, MobileNet V3 was  the most accurate among the tested models for z-line predic-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' From a medical point of view, an ideal model should be  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Performance comparison between different  architecture  Model  ACC  PREC  REC  SPEC  F1  GoogLeNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='846  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='859  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='830  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='862  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='843  MobileNet V3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='842  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='901  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='776  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='908  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='831  ResNet-50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='833  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='865  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='792  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='874  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='826  MobileNet V2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='830  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='852  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='860  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='825  likely to distinguish esophagitis with severe symptoms from  the z-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The average of TP rate were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='950, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='923, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='892, and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='841 for MobileNet V3, MobileNet V2, GoogLeNet, and  ResNet-50, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V3 predicted esophagi-  tis with more confidence than the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' GoogLeNet analysis  Grad-CAM and SHAP were applied to the learned model,  and what kind of the model was created was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1 shows an example of the image results in the case of  TP predicted by GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the Grad-CAM results, red  indicates the most potent activation, and blue indicates the  weakest activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the SHAP results, the SHAP values  of the patches were computed and rendered in a color map:  a positive SHHAP value (red) indicates that the class is sup-  ported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' By contrast, a negative SHAP value (blue) indicates  that the class is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Tearing the esophageal mucosa against the z-line is a  feature of esophagitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1, the results of  Grad-CAM and SHAP showed that the learned model of  GoogLeNet can makes predictions focusing on the clinically  significant aspects of esophagitis images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The GoogLeNet  model learned the findings that are important for diagnosing  esophagitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Comparison results showed that SHAP captured  the location of multiple mucosal tears in the image more ac-  curately than Grad-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2 shows the results of applying Grad-CAM and SHAP  in the FN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The following can be observed from the re-  sults of Grad-CAM and SHAP for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the  Grad-CAM results, most areas in the image are shown as  activated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Areas that provide the basis for the pre-  diction are difficult to identify because of the gradient satu-  ration in the Grad-CAM calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the SHAP results,  the inflammatory areas of the input image are indicated by  blue pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Blue pixels indicate features that have a negative  contribution to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other words, although the  model incorrectly identified esophagitis as a z-line, the model  recognized that areas in the image negatively contributed to  the z-line decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V3 analysis  One hundred images were determined to be TP in the  MobileNet V3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The SHAP results for the images  judged to have the highest and lowest probabilities of being  esophagitis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3, in cases with a high prediction proba-  bility, some features may have a negative contribution to the  (a) Raw image  (b) Grad-CAM  (c) SHAP  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' True Positive Pattern  (a) Raw image  (b) Grad-CAM  (c) SHAP  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' False Negative Pattern  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' First image predicted positive with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='000 probability, and second image predicted positive with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='524 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Many features showing negative contributions  can be identified in the images with low prediction proba-  bility for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In this case, the prediction probability may  be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' CONCLUSIONS  We compared the accuracies of CNN models, including  GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3,  in identifying z-line and esophagitis in endoscopic images  from the open Kvasir dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Among the four models,  GoogLeNet had the highest F1-score, and MobileNet V3  had the highest average TP rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' These results suggest that  GoogLeNet performs better than state-of-the-art CNN mod-  els in medical image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In addition, MoblieNet V3  is a cost-effective model because of its low memory and short  training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Each model was analyzed and compared with  Grad-CAM, and SHAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Other models, datasets, and model  analyses are warranted for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  REFERENCES  [1]  Peng-Jen Chen, Meng-Chiung Lin, Mei-Ju Lai, Jung-  Chun Lin, Henry Horng-Shing Lu, and Vincent S  Tseng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Accurate classification of diminutive colorectal  polyps using computer-aided analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Gastroenterol-  ogy, 154(3):568–575, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [2]  Toshiaki Hirasawa, Kazuharu Aoyama, Tetsuya Tan-  imoto, Soichiro Ishihara, Satoki Shichijo, Tsuyoshi  Ozawa, Tatsuya Ohnishi, Mitsuhiro Fujishiro, Keigo  Matsuo, Junko Fujisaki, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Application of artificial  intelligence using a convolutional neural network for  detecting gastric cancer in endoscopic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Gastric  Cancer, 21(4):653–660, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
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+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='004[3]  Pedro Guimar˜aes, Andreas Keller, Tobias Fehlmann,  Frank Lammert, and Markus Casper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Deep-learning  based detection of gastric precancerous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Gut, 69(1):4–6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [4]  Yaqiong Zhang, Fengxia Li, Fuqiang Yuan, Kai Zhang,  Lijuan Huo, Zichen Dong, Yiming Lang, Yapeng  Zhang, Meihong Wang, Zenghui Gao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Diagnosing  chronic atrophic gastritis by gastroscopy using artificial  intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Digestive and Liver Disease, 52(5):566–  572, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [5]  Konstantin  Pogorelov,  Kristin  Ranheim  Randel,  Carsten Griwodz, Sigrun Losada Eskeland, Thomas  de Lange,  Dag Johansen,  Concetto Spampinato,  Duc-Tien Dang-Nguyen, Mathias Lux, Peter Thelin  Schmidt, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Kvasir: A multi-class image dataset for  computer aided gastrointestinal disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In  Proceedings of the 8th ACM on Multimedia Systems  Conference, pages 164–169, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [6]  Timothy Cogan, Maribeth Cogan, and Lakshman  Tamil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Mapgi:  accurate identification of anatomi-  cal landmarks and diseased tissue in gastrointestinal  tract using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Computers in biology and  medicine, 111:103351, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [7]  Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Ser-  manet, Scott Reed, Dragomir Anguelov, Dumitru Er-  han, Vincent Vanhoucke, and Andrew Rabinovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Go-  ing deeper with convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceedings of the  IEEE conference on computer vision and pattern recog-  nition, pages 1–9, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [8]  Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian  Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In  Proceedings of the IEEE conference on computer vision  and pattern recognition, pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [9]  Mark Sandler, Andrew Howard, Menglong Zhu, An-  drey Zhmoginov, and Liang-Chieh Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Mobilenetv2:  Inverted residuals and linear bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceed-  ings of the IEEE conference on computer vision and  pattern recognition, pages 4510–4520, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [10] Andrew Howard, Mark Sandler, Grace Chu, Liang-  Chieh Chen, Bo Chen, Mingxing Tan, Weijun Wang,  Yukun Zhu, Ruoming Pang, Vijay Vasudevan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Searching for mobilenetv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  In Proceedings of the  IEEE/CVF international conference on computer vi-  sion, pages 1314–1324, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [11] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek  Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv  Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Grad-cam: Visual explanations from deep net-  works via gradient-based localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceedings  of the IEEE international conference on computer vi-  sion, pages 618–626, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [12] Scott M Lundberg and Su-In Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' A unified approach  to interpreting model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Advances in neural  information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [13] Jie Hu, Li Shen, and Gang Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Squeeze-and-excitation  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on  computer vision and pattern recognition, pages 7132–  7141, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}

6tAzT4oBgHgl3EQfEvoZ/content/tmp_files/2301.00997v1.pdf.txt

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+Detector and physics simulation using heavy ion collisions at
+NICA-SPD
+I. Denisenko1,a) and R. Pandey2,b)
+1Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna-141980,
+Moscow Region, Russia.
+2Graduate Engineer Trainee, Larsen & Toubro Limited, Faridabad,
+Haryana, India.
+a)[email protected]
+b)[email protected]
+Abstract
+The space-time picture of hadron formation in high-energy collisions with nuclear
+targets is still poorly known.
+The tests of hadron formation was suggested for the
+first stage of SPD running.
+They will require measuring charged pion and proton
+spectra with the precision better than 10%. A research has been carried out to check
+feasibility of such studies at SPD. In this work, 12C − 12C and 40Ca − 40Ca heavy ion
+collisions at center of mass energy of 11 GeV/nucleon were simulated using the SMASH
+event generator. Firstly, the generator-level events were studied. The distribution of
+track multiplicities and momentum distributions of different types of charged particles
+were obtained. Secondly, the generated events passed through the full reconstruction
+using the SpdRoot framework.
+At this stage particles were identified using dE/dx
+measurement and time-of-flight information. It allowed us to estimate charge track
+multiplicities in the tracking system and purities of charge particles spectra. The results
+on multiplicity are important to estimate occupancies in the tracking system, while the
+results on the pion and proton momentum spectra show that particle identification
+should be acceptable for validation of hadron formation models. This is the first study
+of moderate ion collisions for the SPD Collaboration.
+Keywords:
+Hadron formation effects, Heavy ion collision, SMASH, NICA-SPD, Rapidity,
+Charged track multiplicity, Particle physics event generator.
+1
+arXiv:2301.00997v1  [physics.ins-det]  3 Jan 2023
+
+1
+INTRODUCTION
+The SPD detector is primarily optimized to study spin dependent gluon structure of proton and
+deuteron using open charm production, charmonia production and prompt photons. At the same
+time, its physics program includes studies of various aspects of QCD. The work is devoted to studies
+of hadron formation in nuclear collisions proposed in Ref. [1].
+Hadrons produced in hadron collisions emerge in the form of prehadrons, which interact with
+nucleons with reduced strength. This suppression is poorly known and is described in model de-
+pendent way. This suppression results in different spectra of final particles as is illustrated in Fig.1
+for rapidity distributions (in a similar way it affects the pT spectrum). Naturally, these spectra can
+be used to study hadron formation effects. The required precision of such measurements is 10%.
+The aim of this work is to evaluate feasibility of such measurements with MC simulation. Here,
+ion collisions of 12C − 12C and 40Ca − 40Ca at √s = 11AGeV were generated using the SMASH
+(Simulating Many Accelerated Strongly-interacting Hadrons) event generator. Afterwards, the the
+full simulation and reconstruction was performed using the SpdRoot framework.
+Figure 1: Rapidity spectra of protons and charged pions in 12C − 12C and 40Ca − 40Ca collisions.
+2
+
+12C + 12C, s= 11 GeV
+40Ca + 40Ca, sN = 11 GeV
+105
+106
+Protons
+Protons
+w/oformation
+w/oformation
+default
+default
+QDM
+QDM
+do/dy, mb
+104
+peut = 2 GeV/c -
+ Peut = 2 GeV/c -
+Peut = 1 GeV/c
+Peut = 1 GeV/c
+103
+104
+102
+103
+4 -3 -2 -1
+0
+1
+2
+3
+4
+4
+-3 -2 -1
+0
+1
+2
+3
+4
+y
+y
+12C + 12C, sN= 11 GeV
+40Ca + 40Ca, sNR = 11 GeV
+104
+105
+do/dy, mb
+do/dy, mb
+103
+104
+w/oformation
+w/oformation
+default
+default
+QDM
+QDM
+Peut = 2 GeV/c
+Peut = 2 GeV/c
+Pcut = 1 GeV/c
+pcut = 1 GeV/c
+102
+103
+4 -3 -2 
+-1
+0
+1
+2
+3
+4
+4 -3 -2 -1
+0
+1
+2
+3
+4
+y
+y2
+NICA FACILITY
+The NICA (Nuclotron based Ion Collider fAcility) collider at Joint Institute for Nuclear Research
+in Dubna is is being built to provide beams for two experiments. The first experiment, MPD (Multi
+Purpose Detector), will study properties of dense baryonic matter (matter present at extreme high
+density in QCD phase diagram) like Quark Gluon Plasma. The second experiment, SPD (Spin
+Physics Detector), is devoted to study of spin related phonomena and QCD. Once the NICA collider
+will be operational, scientists will be able to create a special state of matter in laboratory which
+existed for very short interval of time (˜20µ sec) just after the big bang. This special state is called
+as QGP (Quark Gluon Plasma) and it filled the entire universe shortly after the big bang.
+The main parts of NICA facility consists of two independent injector complex (injector for light
+ions, and injector for heavy ions-KRION 6T), Light Ion Linear Accelerator (LU20) for accelerating
+light ions like protons (H+), deutrons, and α-particles upto 5 MeV of K.E, then Heavy Ion Linear
+Accelerator (HILAC) to accelerate heavy ions upto Au to a maximum K.E of 3.2 MeV/n, then a
+Super Conducting (SC) Booster Synchrotron to create ultra high vacuum and to provide complete
+stripping of heavy ions, then a SC Heavy Ion Synchrotron Nuclotron to accelerate both light and
+heavy ions to required beam energy. The accelerated beams will collide at two different locations
+where MPD detector and SPD detector are being built. The schematic view of NICA complex is
+shown in Fig.2.
+Figure 2: Schematic view of NICA complex.
+3
+SPD DETECTOR
+The Spin Physics Detector [2,3] is a 4π universal detector optimized to study spin-related phenomena
+via open charm, charmonia and promopt photons in the collisions of polarized p-p or d-d beams
+with √sNN up to 27 GeV. However, at first stage of NICA-SPD, the expected collision energy
+will be from 3.4 up to 10 GeV, and later on after first upgrade, it is expected to reach upto 27
+GeV. The general layout depicting isometric projection of SPD setup is shown in Fig.3. The main
+parts involved in advanced tracking and particle identification capabilities have been shown. (i)
+The beam pipe passes through the center of the detector, carries the accelerated beams of ions. (ii)
+The MicroMegas detector is to improves the momentum resolution and tracking efficiency of the
+tracking system. (iii) The Straw Tracker (ST) detector is for the reconstruction of the primary and
+3
+
+BM@N Detector
+SPD
+Transport Channel
+HILAC
+Collider
+LU20
+Booster
+-MPD
+Nuclotronsecondary particle tracks and for determination of their momenta. (iv) The Time Of Flight (TOF)
+detector, is a part of Particle Identification (PID) system, and is used for identification of particles
+like π, k, and p with long trajectories. (v) The magnet system shown by red color provides 1T
+of magnetic field along the beam axis. This setup is limited to first stage of SPD operation, and
+will be considered only for the identification of stable charged particles. Neutral particles, like n0,
+photons will be detected at later stages. The main parts of SPD first stage have been explained in
+detail below. There is a possibility to have TOF system for the first stage studies.
+Figure 3: Layout of the SPD setup proposed for first stage at NICA-SPD.
+3.1
+CENTRAL TRACKER
+The innermost detector of SPD consists of a MicroMegas-based Central Tracker (MCT). Its purpose
+is to identify the primary vertex coordinate and to improve momentum resolution and tracking
+efficiency.
+It is based on MicroMegas (Micro Mesh Gaseous Structure) technology and detects
+charged particle by amplifying the charges produced due to ionization of the gas molecules present
+in detector volume. When an ionizing particle track passes through detector volume, it ionizes the
+gas molecules and creates few hundreds of e−-ion pair. Electrons are accelerated opposite to the
+direction of applied electric field of 600 V/cm in ionization gap, while ions are attracted towards
+cathode. When the e− crosses micromesh, it faces intense electric field (> 30 KV/cm) and gains
+enough energy to ionize other gas molecules in its path. During this process an avalanche of e−-ion
+pair is produced (1e− produces 104 e−-ion pairs) which is significant to create an electronic signal
+which is read out by readout electrodes.
+3.2
+STRAW TRACKER
+ST is mainly for the reconstruction of primary and secondary particle tracks and measuring their
+momenta, but also participates in identification of π, K, and p on via energy deposit (dE/dx)
+measurements. It consists of two major parts - barrel (covers radius from 270 to 850 mm) and two
+end-caps. The barrel is divided into 8 modules enclosed in a carbon fiber capsule. Each module has
+30 double layers of straw tubes (dia 1cm) which runs parallel (long straw tubes) and perpendicular
+4
+
+Straw tracker
+Magnet
+Range system
+MicroMegas Endcap
+RangesystemEndcap
+MicroMegas
+Beam-beamcounter
+Beam pipe
+Strawtracker Endcap
+zoomx4
+Zero degree calorimeter(short straw tubes) to the beam axis and contains 1500 and 6000 parallel and perpendicular straw
+tubes respectively. Straw tubes are made of polyethylene terephthalate and outer surface is coated
+with very thin layer of Cu and Au. Carbon capsule is meant to protect the outer surface of these
+tubes from humidity. One side and two opposite ends of capsule are provided with small holes
+where end plugs are fixed. FEE are connected to these end plugs to read the detector signal. Any
+particle which passes through the long straws will send detector signal to both opposite ends while a
+particle passing through short straw will send detector signal to any one side of capsule where FEE
+is attached. Thus, long straws will be read from two opposite ends while short straws will be read
+from one side. The end-caps of ST are divided into 3 modules and each module has 4 hexadecimal
+cameras (U, V, X, Y) to record the four coordinates of any physical quantity like four-momentum.
+The FEE to be used can be similar to the one used at NA64 experiment (for the search of dark
+matter), or DUNE experiment (to detect and study properties of neutrino).
+3.3
+TIME OF FLIGHT DETECTOR
+TOF detector is the part of PID system. Similar to ST, the TOF provides identification of π, k, and
+p by measuring their flight time. The energy loss data registered by ST can be used together with the
+data from TOF for correct identification of particle tracks. The TOF distinguishes charged particles
+(mainly π and k) in the momentum range up to 1.5 GeV. The major parts of TOF comprises of a
+barrel and two end-caps. For the first stage of NICA-SPD, two different designs of TOF has been
+suggested. First one is TOF based on multigap timing Resistive Plate Chambers (mRPC), which
+will consist 220 rectangular plate chambers (160 for the barrel and 30 each for end-caps). Second
+one is based on Plastic Scintillator Tiles and will comprise 10.1K small scintillator tiles (7.4K for
+barrel and 1.4K for each end-caps). Scintillator has a property of emitting light in visible region
+when an ionizing radiation passes through it. So, in this design when a particle passes through
+TOF, scintillated photons are produced which are detected by four Si Photo Multipliers (SiPMs)
+present at each sensor board attached at two extreme ends of scintillator tile.
+4
+EVENT GENERATION
+12C −12C and 40Ca−40Ca heavy ion collisions at √s = 11 AGeV with maximum impact parameter
+set to 8 fm for C-C and 11 fm for Ca-Ca were simulated using SMASH. The fermi motion was
+assumed to be “frozen” and 100K events were generated for each heavy ion collision. The SMASH
+input file for C-C collision is shown below.
+*********** SMASH INPUT ************
+config.yaml file for C-C collision.
+Logging:
+default: INFO
+General:
+Modus:
+Collider
+Time_Step_Mode: Fixed
+Delta_Time:
+0.1
+End_Time:
+200.0
+Randomseed:
+-1
+Nevents:
+100000
+5
+
+Output:
+Output_Interval: 10.0
+Particles:
+Format:
+["Oscar2013"]
+Modi:
+Collider:
+Projectile:
+Particles: {2212: 6, 2112: 6} #C-12
+Target:
+Particles: {2212: 6, 2112: 6} #C-12
+Sqrtsnn: 11.0
+Impact:
+Sample: "quadratic"
+Range: [0.0, 8.0]
+Fermi_Motion: "frozen"
+************************************
+Multiplicity of generated charged particles for C − C and Ca − Ca collisions are shown in
+Fig. 4. The peaks at 12 for 12C+12C collisions and at 40 for 40Ca+40Ca collisions correspond to
+events where no interaction occurred. The rapidity distributions are shown in Fig. 5. The spectra
+obtained from SMASH output show qualitative agreement with the ones in Fig. 1. Peaks for protons
+correspond to particles moving close to the initial beam direction. Moreover, fractions of different
+particle types can be estimated. It can be seen that for |y| < 2 (i.e. within the acceptance of the
+detector) charge particles are dominated by pions. Apart from p±, π±, & K±, marginal numbers
+of sigmas, cascades, and omegas were also generated. The PID efficiency depends on the particle
+momentum.
+The momentum spectra for protons, pions and kaons are shown in Fig. 6 in the
+midrapidity region (|y| < 0.5 for which theoretical predictions has been given) Most of the pions
+have momentum below 0.8 GeV and protons - below 1 GeV. It means that types of these particles
+should be well resolved by dE/dx measurements. When studying pion or proton spectra, there is
+high probability of kaon/pion misidentification, but fraction of such events is strongly suppressed
+by small initial kaon numbers.
+6
+
+(a)
+(b)
+Figure 4: Generator-level multiplicity of charged particles for 12C − 12C collision (a) and 40Ca − 40Ca collisions (b).
+(a)
+(b)
+Figure 5: Rapidity distribution of charged particles in 12C − 12C (a) and 40Ca − 40Ca (b) collision.
+7
+
+Total Multiplicity of Charged Particles, C-12 + C-12
+104
+No. of events
+103
+102
+10
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+No. of charged particlesTotal Multiplicity of Charged Particles, Ca-40 + Ca-40
+104
+No. of events
+103
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+No. of charged particlesRapidity distribution of charged particles, C-12 + C-12
+- protons
+105
+.. pions
+. kaons
+104
+No. of charged particles
+103
+102
+10
+3
+2
+3
+5
+Rapidity of charaed particles (yRapidity distribution of charged particles, Ca-40 + Ca-40
+106
+protons
+pions
+105
+kaons
+No. of charged particles
+104
+103
+102
+10
+5
+3
+2
+Y
+Rapidity of charged particles (y)(a) p distribution of p± in 12C − 12C collision.
+(b) p distribution of p± in 40Ca − 40Ca collision.
+(c) p distribution of π± in 12C − 12C collision.
+(d) p distribution of π± in 40Ca − 40Ca collision.
+(e) p distribution of k± in 12C − 12C collision.
+(f) p distribution of k± in 40Ca − 40Ca collision.
+Figure 6: Total momentum distribution of protons, pions, and kaons at generator level in 12C − 12C and 40Ca− 40Ca collision.
+8
+
+Total momentum distribution of pions, C-12 + C-12
+16000
+14000
+12000
+pions
+10000
+8000
+No.
+6000
+4000
+2000
+0
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+Total momentum of pions (p)Total momentum distribution of pions, Ca-40 + Ca-40
+60000
+50000
+40000
+ON
+30000
+20000
+10000
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+Total momentum of pions (p)Total momentum distribution of kaons, C-12 + C-12
+1000
+800
+ of kaons
+600
+No.
+400
+200
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+Total momentum of kaons (p)Total momentum distribution of kaons, Ca-40 + Ca-40
+4000
+3500
+3000
+kaons
+2500
+2000
+No.
+1500
+1000
+500
+0
+0
+0.2
+0.4
+0.6
+0.8
+1.4
+1.6
+1.8
+2
+Total momentum of kaons (p)Total momentum distribution of protons, C-12 + C-12
+1600
+1400
+1200
+protons
+1000
+800
+ON
+600
+400
+200
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+Total momentum of protons (p)Total momentum distribution of protons, Ca-40 + Ca-40
+6000
+5000
+ of protons
+4000
+No.
+3000
+2000
+1000
+0.2
+0.4
+0.6
+0.8
+1.4
+1.6
+1.8
+2
+2.2
+Total momentum of protons (p)5
+DETECTOR SIMULATION AND EVENT RECONSTRUCTION
+The detector simulation and reconstruction was performed with the SpdRoot framework. To read
+SMASH generated events the SpdRoot code was modified and additional C++ class was added.
+During the simulation stage the particles were transported through the detector geometrical model
+using Geant4.
+At the reconstruction stage, Geant4 tracks and vertices were reconstructed and
+particle identification with dE/dx and time of flight measurements was performed. For the PID
+three hypotheses were considered: pion, kaon and proton. The reconstructed ionization energy
+losses and “measured” time of flight were used to construct conditional probabilities (e.g. p(t|pid),
+where t is the measured time and pid is a particle type hypothesis). Out of 100K events generated
+by SMASH, first 1K events were considered for detector simulation due to slow data processing in
+SpdRoot.
+6
+ANALYSIS
+A physical analysis was performed using C++ codes and ROOT library based on SpdRoot output.
+All tracks reconstructed in the detector with measured momentum were accepted. For the particle
+type the one that gives the largest conditional probability is adopted.
+Multiplicity, as well as
+kinematic distributions for pions, kaons and protons are studied. For particle momentum spectra
+there are no notable differences between C − C and for Ca − Ca collisions, so only the first ones
+will be considered.
+6.1
+CHARGED TRACK MULTIPLICITY
+The SPD detector set-up is optimized for p − p and d − d collisions. Thus knowing charged track
+multiplicities for ion collisions is important to estimate CT and ST occupancies and feasibility of
+such studies. Fig. 7 shows the total multiplicity of charged particles reconstructed by the tracking
+system in 12C − 12C and 40Ca − 40Ca collisions. The numbers of reconstructed tracks are much
+lower compared to generator-level studies. It is because the geometry of the tracking system is such
+that, tracks with polar angle, θ < 10◦ or > 170◦ do not hit the tracker and passes along the beam
+pipe itself, so such tracks are ignored. Also, there were events without nuclei interactions which
+resulted in no track reconstruction. So, to avoid a large peak at zero due to mentioned reasons, the
+X-axis count starts from 1.
+6.2
+PION MOMENTUM SPECTRUM (12C − 12C)
+The spectra of particles identified as pions separately by ionization losses and by TOF are shown in
+Fig. 8 separately. The spectra show resemblance with the generator plot of pion momentum distri-
+bution. Based pn MC-truth information backround from misidentification other charged particles
+(K±, p±, e±, & µ±) is studied. The obtained distribution for “pions identified as pions” only slightly
+deviates from distribution of all selected pion candidates. The estimated relative contamination of
+the pion spectra is shown in Fig. 9. It can seen that purity above 90% can be obtained up to
+1.2 GeV using either dE/dx or TOF measurements.
+9
+
+Figure 7: Charged track multiplicity reconstructed by in 12C − 12C (left), 40Ca − 40Ca (right) collisions (shown by red) and
+number of particles for which TOF information is available (shown by blue).
+(a) Total momentum distribution of reconstructed charged particles
+identified as π± by ionization losses.
+(b) Total momentum distribution of reconstructed charged particles
+identified as π± by TOF.
+Figure 8: Total momentum distribution of reconstructed π± candidates in 12C − 12C collision (Detector level).
+Figure 9: Purity of the selected pion candidates as a function of their momentum.
+10
+
+Total multiplicity of charged particles passing through tracking system, C12-C12
+60
+TOF
+50
+ST
+40
+events
+30
+NO.
+20
+10
+一
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10090
+TOF
+80
+ST
+70
+events
+60
+50
+No.
+40
+30
+20
+10
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100Charged Particles ldentified as Pions by ST, C12-C12
+350
+Pions identified as pions
+Kaons identified as pions
+Protons identified as pions
+300
+Electrons identified as pions
+Muons identified as pions
+250
+Chargedparticlesidentifiedaspions
+200
+150
+100
+50
+0
+.°
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+p(GeV/c)Charged Particles ldentified as Pions by TOF, C12-C12
+Pions identified as pions
+300
+Kaons identified as pions
+Protons identified as pions
+250
+Electrons identified as pions
+Muons identified as pions
+Charged particles identified as pions
+200
+Counts
+150
+100
+50
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+p(GeV/c)Pion spectra precision, C12-C12
+0.8
+0.6
+Counts
+0.4
+0.2
+Precision recordedbyTOF
+Precision recorded by ST
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+p(GeV/c)6.3
+KAON MOMENTUM SPECTRUM (12C − 12C)
+The kaon momentum spectrum was explicitly mentioned among observables to study hadron for-
+mation effects in nuclei. Nevertheless, kaon production may be interesting for the reasons. The
+obtained spectra of kaon candidates is shown in Fig. 10 separately for ionization losses and TOF.
+First of all, the shown data lack statistics. Secondly, it can bee seen that there is a huge contamina-
+tion from misidentified pions. This is explained by very small fraction of generated kaons and the
+fact that probability to select misidentified particle is proportional to their number. The relative
+fraction of correctly identified kaons in shown in Fig. 11.
+(a) Total momentum distribution of reconstructed charged particles
+identified as K± by ionization losses.
+(b) Total momentum distribution of reconstructed charged particles
+identified as K± by TOF.
+Figure 10: Total momentum distribution of reconstructed K± candidates in 12C − 12C collision (Detector level).
+Figure 11: Purity of the selected kaon candidates as a function of their momentum.
+11
+
+Charged Particles ldentified as Kaons by ST, C12-C12
+Kaons identified as kaons
+60
+Pions identified as kaons
+Protons identified as kaons
+Electrons identified as kaons
+50
+Muons identified as kaons
+Charged particles identified askaons
+40
+Counts
+30
+20
+10
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+0
+1.4
+1.6
+1.8
+2
+p(GeV/c)Charged Particles ldentified as Kaons by TOF, C12-C12
+25
+Kaons identified as kaons
+Pions identified as kaons
+Protons identifiedaskaons
+20
+Electrons identified as kaons
+Muons identified as kaons
+Charged particles identified as kaons
+15
+Counts
+10
+5
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+0
+2
+p(GeV/c)Kaon spectra precision, C12-C12
+Precision recorded by TOF
+0.9
+Precision recorded by ST
+0.8
+0.7
+0.6
+unts
+0.5
+Col
+0.4
+0.3
+0.2
+0.1
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+p(GeV/c)6.4
+PROTON MOMENTUM SPECTRUM (12C − 12C)
+Finally, proton momentum spectra have been considered. In this study protons and antiprotons were
+considered together, but the fraction of produced antiprotons is negligible. The proton candidate
+distributions and the contributions from misidentification are shown in Fig. 12. The purity of the
+selected samples is shown in Fig. 13. It can be seen dE/dx measurements alone will not allow
+precise determination of proton spectrum. The reasonably good results can be expected only in
+case of combined identification by ionization losses and TOF system.
+(a) Total momentum distribution of reconstructed charged particles
+identified as p± by ionization losses.
+(b) Total momentum distribution of reconstructed charged particles
+identified as p± by TOF.
+Figure 12: Total momentum distribution of reconstructed p± candidates in 12C − 12C collision (Detector level).
+Figure 13: Purity of the selected proton candidates as a function of their momentum.
+12
+
+Charged Particles ldentified as Protons by ST, C12-C12
+90
+Protons identified as protons
+Kaons identified as protons
+80
+Pions identified as protons
+Electrons identified as protons
+Muons identified as protons
+70
+Charged particles identified as protons
+60
+Counts
+50
+40
+30
+20
+10
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+0
+5
+p(GeV/c)Charged Particles ldentified as Protons by TOF, C12-C12
+90
+Protons identified as protons
+Kaons identified as protons
+80
+Pions identified as protons
+Electrons identified asprotons
+70
+Muons identified as protons
+Charged particles identified as protons
+60
+Counts
+50
+40
+30
+20
+10
+0.5
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+0
+5
+p(GeV/c)Proton spectra precision, C12-C12
+0.8
+Counts
+0.6
+0.4
+0.2
+Precision recorded by TOF
+Precision recorded by ST
+0.5
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+p(GeV/c)7
+SUMMARY
+The goal of this work was to check the feasibility of hadron formation effects studies at the first
+stage of SPD operation. For this purpose an analysis of 12C − 12C and 40Ca − 40Ca collisions were
+performed at the generator level and then the full event reconstruction was done at detector level.
+The multiplicity distributions indicate that occupancies of tracking detectors should be checks.
+Part of the events with high number of charged tracks may not be fully reconstructed. Particle
+identification with ionization losses and TOF was considered separately (for future dE/dx only or
+their combination can be expected). The purity of the measured charged pion distribution for both
+types of ion collisions using dE/dx only is rather good and meets mentioned before requirements. In
+case of combination of information from ionization losses and time of flight system purity of proton
+distribution may be improved.
+References
+[1] V. V. Abramov, A. Aleshko, V. A. Baskov, E. Boos, V. Bunichev, O. D. Dalkarov, R. El-Kholy,
+A. Galoyan, A. V. Guskov and V. T. Kim, et al. Phys. Part. Nucl. 52 (2021) no.6, 1044-1119
+doi:10.1134/S1063779621060022 [arXiv:2102.08477 [hep-ph]].
+[2] V. M. Abazov et al. [SPD proto], [arXiv:2102.00442 [hep-ex]].
+[3] SPD TDR [unpublished].
+13
+

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@@ -0,0 +1,1782 @@
+1
+Federated Coded Matrix Inversion
+Neophytos Charalambidesµ, Mert Pilanciσ, and Alfred O. Hero IIIµ
+.µEECS Department University of Michigan .σEE Department Stanford University
+Email: [email protected], [email protected], [email protected]
+Abstract
+Federated learning (FL) is a decentralized model for training data distributed across client devices. Coded computing (CC) is
+a method for mitigating straggling workers in a centralized computing network, by using erasure-coding techniques. In this work
+we propose approximating the inverse of a data matrix, where the data is generated by clients; similar to the FL paradigm, while
+also being resilient to stragglers. To do so, we propose a CC method based on gradient coding. We modify this method so that the
+coordinator does not need to have access to the local data, the network we consider is not centralized, and the communications which
+take place are secure against potential eavesdroppers.
+I. INTRODUCTION AND RELATED WORK
+Inverting a matrix is one of the most important operations in numerous applications, such as, signal processing, machine
+learning, and scientific computing [2], [3]. A common way of inverting a matrix is to perform Gaussian elimination, which
+requires O(N 3) operations for square matrices of order N. In high-dimensional applications, this can be cumbersome. Over the
+past few years the machine learning (ML) community has made much progress on federated learning (FL), focusing on iterative
+methods.
+The objective of FL is to leverage computation, communication and storage resources to perform distributed computations for
+ML models, where the data of each federated worker is never shared with the coordinator of the network; that aggregates local
+computations in order to update the model parameters. In FL applications it is important that the data is kept private and secure.
+Distributed computations in the presence of stragglers (workers who fail to compute their task or have longer response time than
+others) must account for the effect of non-responsive workers. Coding-theoretic approaches have been adopted for this purpose
+[4], [5], and fall under the framework of coded computing (CC). Data security is also an increasingly important issue in CC [6].
+Despite the fact that multiplication algorithms imply inversion algorithms and vice versa, in the context of CC; matrix inversion
+has not been studied as extensively as coded matrix multiplication (CMM) [7]. The main reason for this is the fact that the latter
+is non-linear and non-parallelizable as an operator. We point out that distributed inversion algorithms do exist, though these make
+assumptions on the matrix, are specific for distributed and parallel computing platforms, and require a matrix factorization; or
+heavy and multiple communication instances between the servers and the coordinator.
+In [1] a CC method1 was proposed based on gradient coding (GC) [8], which approximates the inverse of a matrix A. In order
+to overcome the obstacle of non-linearity, the columns of A−1 are approximated. When assuming floating-point arithmetic, this
+CCM introduces no numerical nor approximation errors. Note that GC and not CMM was utilized, as the latter does not require
+the encoding to be done locally by the workers.
+Though the two areas of FL and CC seem to be closely related, on the surface they appear incompatible. For instance, in CC
+one often assumes there is a master server that distributes the data and may perform the encoding (encoding by the master server
+is done in CMM, but not in GC), while in FL the central coordinator never has access to the distributed local training data; which
+are located at different client nodes or workers.
+There are a few recent works that leverage CC in order to devise secure FL methods for distributed regression and iterative
+optimization [9]–[14]. In this work, we combine optimization and CC, using erasure coding to protect against stragglers as in
+CC and locally approximating the inverse without revealing the data to the coordinator, to design a FL scheme. Our approach,
+is based on the coded matrix inversion method (CMIM) we develop, which utilizes balanced Reed-Solomon (BRS) codes [15],
+[16]. This results in an efficient decoding in terms of the threshold number of responsive workers needed to perform an error free
+computation. We show that the general class of maximum distance separable (MDS) generator matrices could be used to generate
+a suitable erasure code (Theorem 6). The focus is on BRS codes, which have the following advantages:
+(i) minimum redundancy per job across the network,
+(ii) they optimize communication from workers to the master,
+(iii) we can efficiently decode the resulting method.
+Our CMIM can also be used to compute the Moore–Penrose pseudoinverse Y† of a data matrix Y ∈ RM×N for M ≫ N,
+which is more general than inverting a square matrix. By using the fact that Y† = (Y⊤Y)−1Y⊤, the bottleneck is computing
+the inverse of A = Y⊤Y. In addition, two more matrix multiplications need to take place distributively: computing A before
+the inversion; and �
+A−1Y⊤ after the inverse has been approximated. The matrix products can be computed distributively using
+A preliminary version also considers approximating A† [1], in the CC setting. This work was partially supported by grants ARO W911NF-15-1-0479 and Dept
+of Energy DE-NA0003921.
+1We abbreviate ‘coded computing method/methods’ to CCM/CCMs.
+arXiv:2301.03539v1  [cs.IT]  9 Jan 2023
+
+2
+various CCMS, e.g. we can use a modification of the coded FL approaches of [11] and a CMM from [17]; both of which are based
+on GC. For the remainder of the paper, we focus on the generic problem of inverting a square matrix A.
+The proposed FL approach applies to general linear regression problems. Compared to previous FL iterative approaches [18],
+the difference is that for Yθ = p; with p the label vector and θ the model parameters, the pseudoinverse-regularized regression
+solution is ˆθ = �
+Y†p. Unlike conventional FL methods, this regularized regression can be computed non-iteratively. The non-
+iterative nature of the proposed approach is advantageous in settings such as Kalman filtering, where the matrix inverse must be
+updated in real time as measurements come in.
+The paper is organized as follows. In II we recall basic facts on matrix inversion, least squares approximation and finite fields.
+In III we review BRS codes, and prove two key lemmas regarding their generator matrices. In IV we present the matrix inverse
+approximation algorithm we utilize in our CCM. The main contribution is presented in V. Our federated approach is split into four
+phases, which we group in pairs of two. First, we discuss information sharing from the coordinator to the workers (we consider
+all the clients’ servers as the network’s workers), and then information sharing between the workers. Second, we show how our
+inversion algorithm can be incorporated in linear CCMs, and describe how this fits into the FL paradigm. Concluding remarks and
+future work are presented in VI.
+A. Overview of the Coded Matrix Inversion Method
+In CC the computational network is centralized, and is comprised of a master server who communicates with n workers. The
+idea behind our approximation is that the workers use a least squares solver to approximate multiple columns of A−1, resulting
+in a set of local approximations to submatrices of �
+A−1, which we refer to as blocks. We present approximation guarantees and
+simulation results for steepest descent (SD) and conjugate gradient (CG) iterative optimization methods. By locally approximating
+the columns in this way, the workers can linearly encode the blocks of �
+A−1. The clients have a block of data {Aι}k
+ι=1, which
+constitute the data matrix A =
+�
+A1 · · · Ak
+�
+. To simplify our presentation, we assume that each local data block is of the same
+size; i.e. Aι ∈ RN×T for T = N/k, and that client i has ni servers. Therefore, the total number of servers is n = �k
+j=1 nj.
+We assume the blocks are of the same size, so that the encodings carried out by the clients are consistent. In V, we show that this
+assumption is not necessary. Moreover, for the CCM, it is not required that the number of blocks equal the number of clients. For
+a given natural number γ, assume that γ divides T; denoted γ | T (each local data block Aι is further divided into γ sub-blocks).
+In the case where k ∤ N or γ ∤ T, we can pad the blocks of �
+A−1 so that these assumptions are met.
+A limitation of our proposed CMIM, is the fact that each server needs to have full knowledge of A, in order to estimate columns
+of A−1 through a least squares solver. The sensitivity of Gaussian elimination and matrix inversion also requires that all clients
+have knowledge of each others’ data [1]. This limitation is shared by other coded federated learning methods, e.g. CodedPaddedFL
+[11]. In contrast to CC and GC; where a master server has access to all the data, in FL the data is inherently distributed across
+devices, thus GC cannot be applied directly. We also assume that the coordinator does not intercept the communication between
+the clients, otherwise she could recover the local data. Also, we trust that the coordinator will not invert �
+A−1, to approximate A
+— this would be computationally difficult, for N large.
+Before broadcasting the data amongst themselves, the clients encode their block Ai, which guarantees security from outside
+eavesdroppers. When the clients receive the encoded data, they can decrypt and recover A. Then, their servers act as the workers
+of the proposed CMIM and carry out their assigned computations, and directly communicate their computations back to the
+coordinator. Once the recovery threshold (the minimum number of responses needed to recover the computation) is met, the
+approximation �
+A−1 is recoverable.
+B. Coded Federated Learning
+There are few works that leverage CC to devise secure FL schemes. Most of these works have focused on distributed regression
+and iterative methods, which is the primary application for FL [9]–[13]. Below, we describe and compare these approaches to our
+work.
+The authors of [9] proposed coded federated learning, in which they utilize a CMM scheme. Their security relies on the use
+of random linear codes, to define the parity data. Computations are carried out locally on the systematic data, and only the parity
+data is sent to the coordinator. The main drawback compared to our scheme is that each worker has to generate a random encoding
+matrix and apply a matrix multiplication for the encodings, while we use the same BRS generator matrix across the network,
+based on GC, to linearly encode the local computations. The drawback in our case, is that the workers need to securely share their
+data with each other. This is an artifact of the operation (inversion) we are approximating, and is inevitable in the general case
+where A has no structure. Under the FL setting we are considering, where the data is gathered or generated locally and is not
+i.i.d., we cannot make any assumptions on the structure of A.
+In [11], two methods were proposed. CodedPaddedFL combines one-time-padding with GC to carry out the FL task. Some
+of its disadvantages are that a one-time-pad (OTP) needs to be generated by each worker, and that the OTPs are shared with the
+coordinator, which means that if she gets hold of the encrypted data, she can decrypt it, compromising security. Furthermore,
+there is a heavy communication load and the coordinator needs to store all the pads in order to recover the computed gradients. In
+
+3
+the proposed CMIM, the coordinator generates a set of interpolation points, and shares them with the clients. If the coordinator
+can intercept the communication between the workers, she can decrypt the encrypted data blocks. The second method proposed
+in [11], CodedSecAgg, relies on Shamir’s secret sharing (SSS); which is based on polynomial interpolation over finite fields. In
+contrast, our CMIM relies on GC and Lagrange interpolation.
+Lastly, we discuss the method proposed in [13], which is based on the McEliece cryptosystem, and moderate-density parity-
+check codes. This scheme considers a communication delay model which defines stragglers as the workers who respond slower
+than the fastest worker, and time out after a predetermined amount of time ∆. As the iterative SD process carries on, such workers
+are continuously disregarded. Due to this, there is a data sharing step at each iteration, at which the new stragglers communicate
+encrypted versions of their data to the active workers. Our scheme is non-iterative, and has a fixed recovery threshold. Unlike
+some of the works previously mentioned, which guarantee information-theoretic security, the McEliece based systems and our
+approach have computational privacy guarantees.
+C. Lagrange Interpolation Coded Computing Methods
+While there is extensive literature on matrix-vector and matrix-matrix multiplication, and computing the gradient in the presence
+of stragglers, there is limited work on computing or approximating the inverse of a matrix [19]. The non-linearity of matrix
+inversion prohibits linear or polynomial encoding of the data before the computations are to be performed. Consequently, most
+CCMs cannot be directly utilized. GC is the appropriate CC set up to consider [20], precisely because the encoding takes place
+once the computation has been completed, in contrast to most CMM methods where the encoding is done by the master, before
+the data is distributed.
+Here, we give a brief overview of the GC on which our CMIM is based. We also review “Lagrange Coded Computing” (LCC),
+which has relations to our approach. Then, we give a summary of our proposed CMIM. All these rely on Lagrange interpolation
+over finite fields.
+Gradient codes are a class of codes designed to mitigate the effect of stragglers in data centers, by recovering the gradient of
+differentiable and additively separable objective functions in distributed first order methods [20]. The proposed CMIM utilizes
+BRS generator matrices constructed for GC [8]. The main difference from our work is that in GC the objective is to construct an
+encoding matrix G and decoding vectors aI ∈ Ck, such that a⊤
+I G = ⃗1 for any set of non-straggling workers indexed by I. To
+do so, the decomposition of the BRS generator matrices GI = HIP is exploited, where HI is a Vandermonde matrix; and the
+first row of P is equal to ⃗1. Subsequently a⊤
+I is extracted as the first row of H−1
+I .
+In the proposed CMIM framework, the objective is to design an encoding-decoding pair ( ˜G, ˜DI) for which ˜DI ˜G = IN, for
+all I ⊊ Nn of size k. The essential reason for requiring this condition, as opposed to that of GC, is that the empirical gradient of a
+given dataset is the sum of each individual gradients, while in our scenario if the columns of �
+A−1 are summed; they cannot then
+be recovered.
+The state-of-the art CC framework is LCC, which is used to compute arbitrary multivariate polynomials of a given dataset
+[5], [21]. This approach is based on Lagrange interpolation, and it achieves the optimal trade-off between resiliency, security,
+and privacy. The problem we are considering is not a multivariate polynomial in terms of A. To securely communicate A to
+the workers, we encode it through Lagrange interpolation. Though similar ideas appear in LCC, the purpose and application of
+the interpolation is different. Furthermore, LCC is a point-based approach [22] and requires additional interpolation and linear
+combination steps after the decoding takes place.
+Recall that the workers in the CMIM must compute blocks of �
+A−1. Once they complete their computations, they encode them
+by computing a linear combination with coefficients determined by a sparsest-balanced MDS generator matrix. Referring to the
+advantages claimed for CMIM in Section I, working with MDS generator matrices allows us to meet points (i) and (ii), while BRS
+generator matrices also help us satisfy (iii). Once the recovery threshold is met, the coordinator can recover the approximation
+�
+A−1. The structure of sparsest-balanced generator matrices is also leveraged to optimally allocate tasks to the workers, while
+linear encoding is what allows minimal communication load from the workers to the master. Security against eavesdroppers is
+guaranteed by encoding the local data through a modified Lagrange interpolation polynomial, before it is shared by the clients.
+This CMIM also extends to approximating A† [1].
+II. PRELIMINARY BACKGROUND
+The set of N ×N non-singular matrices is denoted by GLN(R). Recall that A ∈ GLN(R) has a unique inverse A−1, such that
+AA−1 = A−1A = IN. The simplest way of computing A−1 is by performing Gaussian elimination on
+�
+A|IN
+�
+, which gives
+�
+IN
+��A−1] in O(N 3) operations. In Algorithm 1, we approximate A−1 column-by-column. We denote the ith row and column
+of A respectively by A(i) and A(i). The condition number of A is κ2 = ∥A∥2∥A−1∥2. For I an index subset of the rows of a
+matrix M, the matrix consisting only of the rows indexed by I, is denoted by MI.
+In the proposed algorithm we approximate N instances of the least squares minimization problem
+θ⋆
+ls = arg min
+θ∈RM
+�
+∥Aθ − y∥2
+2
+�
+(1)
+
+4
+for A ∈ RN×M and y ∈ RN. In many applications N ≫ M, where the rows represent the feature vectors of a dataset. This has
+the closed-form solution θ⋆
+ls = A†y.
+Computing A† to solve (1) is intractable for large M, as it requires computing the inverse of A⊤A. Instead, we use gradient
+methods to get approximate solutions, e.g. SD or CG, which require less operations, and can be done distributively. One could use
+second-order methods; e.g. Newton–Raphson, Gauss-Newton, Quasi-Newton, BFGS, or Krylov subspace methods instead. This
+would be worthwhile future work.
+When considering a minimization problem with a convex differentiable objective function ψ: Θ → R over an open constrained
+set Θ ⊆ RM, as in (1), the SD procedure selects an initial θ[0] ∈ Θ, and then updates θ according to:
+θ[t+1] = θ[t] − ξt · ∇θψ(θ[t])
+for t = 1, 2, 3, ...
+until a termination criterion is met, for ξt the step-size. The CG method is the most used and prominent iterative procedure for
+numerically solving systems of positive-definite equations.
+Our proposed coding scheme is defined over the finite field of q elements, Fq. We denote its cyclic multiplicative subgroup
+by F×
+q = Fq\{0Fq}. For implementation purposes, we identify finite fields with their realization in C as a subgroup of the circle
+group, since we assume our data is over R. All operations can therefore be carried out over C. Specifically, for β ∈ F×
+q a generator,
+we identify βj with e2πij/q, and 0Fq with 1. The set of integers between 1 and ν is denoted by Nν.
+III. BALANCED REED-SOLOMON CODES
+A Reed-Solomon code RSq[n, k] over Fq for q > n > k, is the encoding of polynomials of degree at most k − 1, for k the
+message length and n the code length. It represents our message over the defining set of points A = {αi}n
+i=1 ⊂ Fq
+RSq[n, k] =
+��
+f(α1), f(α2), · · · , f(αn)
+� ���
+f(X) ∈ Fq[X] of degree ⩽ k − 1
+�
+where αi = αi, for α a primitive root of Fq. Hence, each αi is distinct. A natural interpretation of RSq[n, k] is through its encoding
+map. Each message (m0, ..., mk−1) ∈ Fk
+q is interpreted as f(x) = �k−1
+i=0 mixi ∈ Fq[x], and f is evaluated at each point of A.
+From this, RSq[n, k] can be defined through the generator matrix
+G =
+�
+�
+�
+�
+�
+1
+α1
+α2
+1
+. . .
+αk−1
+1
+1
+α2
+α2
+2
+. . .
+αk−1
+2
+...
+...
+...
+...
+...
+1
+αn
+α2
+n
+. . .
+αk−1
+n
+�
+�
+�
+�
+� ∈ Fn×k
+q
+,
+thus, RS codes are linear codes over Fq. Furthermore, they attain the Singleton bound, i.e. d = n − k + 1, where d is the code’s
+distance, which implies that they are MDS.
+Balanced Reed-Solomon codes [15], [16] are a family of linear MDS error-correcting codes with generator matrices G ∈ Fn×k
+q
+that are:
+• sparsest: each column has the least possible number of nonzero entries
+• balanced: each row contains the same number of nonzero entries
+for the given code parameters k and n. The design of these generators are suitable for our purposes, as:
+1) we have balanced loads across homogeneous workers,
+2) sparse generator matrices reduce the computation tasks across the network,
+3) the MDS property permits an efficient decoding step,
+4) linear codes produce a compressed representation of the encoded blocks.
+A. Balanced Reed-Solomon Codes for CC
+In the proposed CMIM, we leverage BRS generator matrices to approximate A−1. For simplicity, we will consider the case
+where d = s + 1 = nw
+k is a positive integer2, for n the number of workers and s the number of stragglers. Furthermore, d is
+the distance of the code and ∥G(j)∥0 = d for all j ∈ Nk; ∥G(i)∥0 = w for all i ∈ Nn, and d > w since n > k. For decoding
+purposes, we require that at least k = n − s workers respond. Consequently, d = s + 1 implies that n − d = k − 1. For simplicity,
+we also assume d ⩾ n/2. In our setting, each column of G corresponds to a computation task of �
+A−1; which we will denote by
+ˆ
+Ai, and each row corresponds to a worker.
+2The case where nw
+k
+∈ Q+\Z+ is analysed in [8], and also applies to our approach. We restrict our discussion to the case where nw
+k
+∈ Z+.
+
+5
+Our choice of such a generator matrix G ∈ Fn×k
+q
+, solves
+arg
+min
+G∈Fn×k
+q
+�
+nnzr(G)
+�
+s.t.
+∥G(i)∥0 = w, ∀i ∈ Nn
+∥G(j)∥0 = d, ∀i ∈ Nk
+rank(GI) = k, ∀I : |I| = k
+(2)
+which determines an optimal task allocation among the workers of the proposed CMIM.
+Under the above assumptions, the entries of the generator matrix of a BRSq[n, k] code meet the following:
+• each column is sparsest, with exactly d nonzero entries
+• each row is balanced, with w = dk
+n nonzero entries
+where d equals to the number of workers who are tasked to compute each block, and w is the number of blocks that are computed
+by each worker.
+Each column G(j) corresponds to a polynomial pj(x), whose entries are the evaluation of the polynomial we define in (3) at
+each of the points of the defining set A, i.e. Gij = pj(αi) for (i, j) ∈ Nn × Nk. To construct the polynomials {pj(x)}k
+j=1, for
+which deg(pj) ⩽ nnzr(G(j)) = n − d = k − 1, we first need to determine a sparsest and balanced mask matrix M ∈ {0, 1}n×k,
+which is ρ-sparse for ρ = d
+n; i.e. nnzr(G) = ρnk. We use the construction from [8], though it is fairly easy to construct more
+general such matrices, by using the Gale-Ryser Theorem [23], [24]. Furthermore, deterministic constructions resemble generator
+matrices of cyclic codes.
+For our purposes we use B as our defining set of points, where each point corresponds to the worker with the same index. The
+objective now is to devise the polynomials pj(x), for which pj(βi) = 0 if and only if Mij = 0. Therefore:
+(I) Mij = 0
+=⇒
+(x − βi) | pj(x)
+(II) Mij ̸= 0
+=⇒
+pj(βi) ∈ F×
+q
+for all pairs (i, j).
+The construction of BRS[n, k]q from [15] is based on what the authors called scaled polynomials. Below, we summarize the
+polynomial construction based on Lagrange interpolation [8]. We then prove a simple but important result that allows us to
+efficiently perform the decoding step.
+The univariate polynomials corresponding to each column G(j), are defined as:
+pj(x) :=
+�
+i:Mij=0
+� x − βi
+βj − βi
+�
+=
+k
+�
+ι=1
+pj,ι · xι−1 ∈ Fq[x]
+(3)
+which satisfy (I) and (II). By the BCH bound [25, Chapter 9], it follows that deg(pj) ⩾ n − d = k − 1 for all j ∈ Nk. Since
+each pj(x) is the product of n − d monomials, we conclude that the bound on the degree is satisfied and met with equality, hence
+pj,ι ∈ F×
+q for all coefficients.
+By construction, both G and GI are decomposable into a Vandermonde matrix H ∈ Bn×k and a matrix comprised of the
+polynomial coefficients H ∈ (F×
+q )k×k [8]. Specifically, G = HP where Hij = βj−1
+i
+= βi(j−1) and Pij = pj,i are the
+coefficients from (3). This can be interpreted as P(j) defining the polynomial pj(x), and H(i) is comprised of the first k positive
+powers of βi in ascending order, therefore
+pj(βi) =
+k
+�
+ι=1
+pj,ι · βι−1
+i
+= ⟨H(i), P(j)⟩.
+The following lemmas will help us respectively establish in our CC setting the efficiency of our decoding step and the optimality
+of the allocated tasks to the workers. For Lemma 1, recall that efficient matrix multiplication algorithms have complexity O(N ω),
+for ω < 2.373 the matrix multiplication exponent [26].
+Lemma 1. The restriction GI ∈ Fk×k
+q
+of G to any of its k rows indexed by I ⊊ Nn, is an invertible matrix. Moreover, its inverse
+can be computed online in O(k2 + kω) operations.
+Proof. The matrices H and P are of size n × k and k × k respectively. The restricted matrix GI is then equal to HIP, where
+HI ∈ Fk×k
+q
+is a square Vandermonde matrix, which is invertible in O(k2) time [27]. Specifically
+HI =
+�
+�
+�
+�
+�
+1
+βI1
+β2
+I1
+. . .
+βk−1
+I1
+1
+βI2
+β2
+I2
+. . .
+βk−1
+I2
+...
+...
+...
+...
+...
+1
+βIk
+β2
+Ik
+. . .
+βk−1
+Ik
+�
+�
+�
+�
+� ∈ Fk×k
+q
+.
+
+6
+It follows that
+det(HI) =
+�
+{i<j}⊆I
+(βj − βi)
+which is nonzero, since β is primitive. Therefore, HI is invertible. By [15, Lemma 1] and the BCH bound, we conclude that P is
+also invertible. Hence, GI is invertible for any set I.
+Note that the inverse of P can computed a priori by the master before we deploy our CCM. Therefore, computing G−1
+I
+online
+with knowledge of P−1, requires an inversion of HI which takes O(k2) operations; and then multiplying it by P−1. Thus, it
+requires O(k2 + kω) operations in total.
+■
+Lemma 2. The generator matrix G ∈ Fn×k
+q
+of a BRSq[n, k] MDS code defined by the polynomials pj(x) of (3), solves the
+optimization problem (2).
+Proof. The first two constraints are satisfied by the definition of G, which meets the sparsest and balanced constraints with
+equality; for the given parameters. The last constraint is implied by the MDS theorem, which states that every set of k rows of G
+is linearly independent.
+The sparsity constraints of (2) imply that nnzr(G) ⩾ max{nw, kd}, and for our parameters we have nw = kd. This condition
+is met with equality for the chosen G, as
+nnzr(G) =
+�
+j∈Nk
+nnzr(G(j))
+=
+�
+j∈Nk
+#
+�
+pj(βi) ̸= 0 : βi ∈ B
+�
+=
+�
+j∈Nk
+n −
+�
+i : Mij = 0
+�
+=
+�
+j∈Nk
+n − (n − d)
+= kd
+and the proof is complete.
+■
+IV. INVERSE APPROXIMATION ALGORITHM
+Our goal is to estimate A−1 =
+�
+b1 · · · bN
+�
+, for A a square matrix of order N. A key property to note is
+AA−1 = A
+�
+b1 · · · bN
+�
+=
+�
+Ab1 · · · AbN
+�
+= IN
+which implies that Abi = ei for all i ∈ NN, where ei are the standard basis column vectors. Assume for now that we use any
+black-box least squares solver to estimate
+ˆbi ≈ arg min
+b∈RN
+�
+fi(b) := ∥Ab − ei∥2
+2
+�
+(4)
+which we call N times, to recover �
+A−1 :=
+�ˆb1 · · · ˆbN
+�
+. This approach may be viewed as approximating
+�
+A−1 ≈ arg
+min
+B∈RN×N
+�
+∥AB − IN∥2
+F
+�
+.
+Alternatively, one could estimate the rows of A−1. Algorithm 1 shows how this can be performed by a single server.
+Algorithm 1: Estimating A−1
+Input: A ∈ GLN(R)
+for i=1 to N do
+approximate ˆbi ≈ arg minb∈RN
+�
+∥Ab − ei∥2
+2
+�
+end
+return �
+A−1 ←
+�ˆb1 · · · ˆbN
+�
+In the case where SD is used to approximate ˆbi from (4), the overall operation count is O(TiN 2); for Ti the total number of
+descent iterations used. An upper bound on the number of iterations can be determined by the underlying termination criterion,
+e.g. the criterion fi(ˆb[t]) − fi(b⋆
+ls) ⩽ ϵ is guaranteed to be satisfied after T = O(log(1/ϵ)) iterations [28]. The overall error of
+�
+A−1 may be quantified as
+
+7
+• errℓ2( �
+A−1) := ∥ �
+A−1 − A−1∥2
+• errF ( �
+A−1) := ∥ �
+A−1 − A−1∥F
+• errrF ( �
+A−1) := ∥ �
+A−1−A−1∥F
+∥A−1∥F
+=
+N
+�
+i=1
+∥Aˆbi−ei∥2
+∥A−1∥F
+.
+To approximate A−1 distributively, each of the n servers are asked to estimate τ-many ˆbi’s in parallel. When using SD, the
+worst-case runtime by the workers is O(τTmaxN 2), for Tmax the maximum number of iterations of SD among the workers. If CG
+is used, each worker needs no more than a total of Nτ CG steps to exactly compute its task, i.e. O(τNκ2) operations; as each
+instance of (4) is expected to converge in O(κ2) iterations, which is the worst case runtime [29], [30].
+In order to bound errrF ( �
+A−1) = ∥ �
+A−1−A−1∥F
+∥A−1∥F
+, we first upper bound the numerator and then lower bound the denominator.
+Since ∥A−1 − �
+A−1∥2
+F = �N
+i=1 ∥A−1ei − ˆbi∥2
+2, bounding the numerator reduces to bounding ∥A−1ei − ˆbi∥2
+2 for all i ∈ NN.
+This is straightforward
+∥A−1ei − ˆbi∥2
+2
+♦
+⩽ 2
+�
+∥A−1ei∥2
+2 + ∥ˆbi∥2
+2
+�
+$
+⩽ 2
+�
+∥A−1∥2
+2 · ∥ei∥2
+2 + ∥ˆbi∥2
+2
+�
+= 2
+�
+1/σmin(A)2 + ∥ˆbi∥2
+2
+�
+(5)
+where in ♦ we use the fact that ∥u − v∥2
+2 ⩽ 2(∥u∥2
+2 + ∥v∥2
+2), and in $ the submultiplicativity of the ℓ2-norm is invoked. For the
+denominator, by the definition of the Frobenius norm
+∥A−1∥2
+F =
+N
+�
+i=1
+1
+σi(A)2 ⩾
+N
+σmax(A)2 .
+(6)
+By combining (5) and (6) we get
+errrF ( �
+A−1) ⩽
+√
+2
+�
+N/σmin(A)2 + �N
+i=1 ∥ˆbi∥2
+2
+N/σmax(A)2
+�1/2
+=
+√
+2
+�
+κ2
+2 + σmax(A)2
+N
+·
+N
+�
+i=1
+∥ˆbi∥2
+2
+�1/2
+.
+This is an additive error bound in terms of the problem’s condition number, which also shows a dependency on the estimates
+{ˆbi}N
+i=1. Propositions 3 and 4 give error bounds when using SD and CG as the subroutine of Algorithm 1 respectively.
+Proposition 3. For A ∈ GLN(R), we have errF ( �
+A−1) ⩽
+ϵ√
+N/2
+σmin(A)2 and errrF ( �
+A−1) ⩽
+ϵ√
+N/2
+σmin(A), when using SD to solve (4) with
+termination criteria ∥∇fi(b[t])∥2 ⩽ ϵ for each i.
+Proof. Recall that for a strongly-convex function with strong-convexity parameter µ, we have the following optimization gap [28,
+Section 9.1.2]
+fi(b) − fi(b⋆
+ls) ⩽ 1
+2µ · ∥∇fi(b)∥2
+2 .
+(7)
+For A ∈ GLN(R) in (4), the constant is µ = σmin(A)2. By fixing ϵ =
+�
+2σmin(A)2η, we have η = 1
+2 ·
+�
+ϵ
+σmin(A)
+�2
+. Thus, by (7)
+and our termination criterion:
+∥∇fi(b)∥2 ⩽
+�
+2σmin(A)2η
+=⇒
+fi(b) − fi(b⋆
+ls) ⩽ η ,
+so when solving (4) we get
+fi(b) − fi(b⋆
+ls) = fi(b) − 0 = ∥Aˆbi − ei∥2
+2 ,
+hence
+∥Aˆbi − ei∥2
+2 ⩽ 1
+2 ·
+�
+ϵ
+σmin(A)
+�2
+(8)
+
+8
+for all i ∈ NN. We want an upper bound for each summand ∥A−1ei − ˆbi∥2
+2 of the numerator of errrF ( �
+A−1)2:
+∥A−1ei − ˆbi∥2
+2 = ∥A−1(ei − Aˆbi)∥2
+2
+⩽ ∥A−1∥2
+2 · ∥ei − Aˆbi∥2
+2
+♯
+⩽ ∥A−1∥2
+2 · 1
+2 ·
+�
+ϵ
+σmin(A)
+�2
+(9)
+=
+ϵ2
+2σmin(A)4
+(10)
+where ♯ follows from (8), thus errF ( �
+A−1)2 ⩽
+Nϵ2
+2σmin(A)4 . Substituting (9) into the definition of errrF ( �
+A−1) gives us
+errrF ( �
+A−1)2 ⩽ ∥A−1∥2
+2
+∥A−1∥2
+F
+· N
+2 ·
+�
+ϵ
+σmin(A)
+�2 ‡
+⩽
+Nϵ2/2
+σmin(A)2
+where ‡ follows from the fact that ∥A−1∥2
+2 ⩽ ∥A−1∥2
+F .
+■
+In the experiments of Subsection IV-A, we verify that Proposition 3 holds for Gaussian random matrices. The dependence on
+1/σmin(A) is an artifact of using gradient methods to solve the underlying problems (4), since the error will be multiplied by
+∥A−1∥2
+2. In theory, this can be annihilated if one runs the algorithm on pA for p ≈ 1/σmin(A), followed by multiplication of
+the final result by p. This is a way of preconditioning SD. In practice, the scalar p should not be selected to be much larger than
+1/σmin(A), as it could result in �
+A−1 ≈ 0N×N.
+Proposition 4. Assume Algorithm 1 uses CG to solve (4). Then, in O
+�
+N√κ2 ln(1/ϵ)
+�
+iterations, we have errF ( �
+A−1) ⩽ Nϵ.
+Moreover, if A⊤A has ˜N distinct eigenvalues, it converges in at most ˜NN steps.
+Proof. By [29, Section 10] and [31, Section 2], we know that for each subroutine (4) of Algorithm 3, CG requires at most
+O(√κ2 ln(1/ϵ)) iterations in order to attain an ϵ-optimal point, for each ˆbi. Hence, considering all approximate columns {ˆbi}N
+i=1,
+we conclude that the total error in terms of the Frobenius norm of �
+A−1, is at most Nϵ.
+Recall that in order to solve (1) with the CG method in the case where A is neither symmetric, positive-definite, nor square, we
+apply the CG iteration to the normal equations
+A⊤Ay = A⊤θ .
+This follows by setting the derivative of (1) to zero. In our scenario, we are assuming that A ∈ GLN(R), hence A⊤A is full-rank
+and symmetric, thus CG in its simplest form can be used to solve the minimization problems of Algorithm 1. By [30, Theorem
+38.4], it follows that each instance of (4) converges in at most ˜N steps.
+■
+Even though Proposition 4 guarantees convergence in at most ˜NN steps, it does not assume floating-point arithmetic. Therefore,
+this does not hold in practical settings. Our experiments though show that after significantly less steps, we achieve approximations
+of negligible error, which is sufficient for ML and FL applications.
+A. Numerical Experiments
+The accuracy of the proposed algorithms was tested on randomly generated matrices, using both SD and CG for the subroutine
+optimization problems. The depicted results are averages of 20 runs, with termination criteria ∥∇fi(b[t])∥2 ⩽ ϵ for SD and
+∥b[t]
+i − b[t−1]
+i
+∥2 ⩽ ϵ for CG, for the given ϵ accuracy parameters. We considered A ∈ R100×100. The error subscripts represent
+A = {ℓ2, F, rF}, N = {ℓ2, F}. We note that significantly fewer iterations took place when CG was used for the same ϵ, though
+this depends heavily on the choice of the step-size. The errors observed in the case of CG, are due to floating-point arithmetic.
+Therefore, there is a trade-off between accuracy and speed when using SD vs. CG.
+Average �
+A−1 errors, for A ∼ 50 · N(0, 1) — SD
+ϵ
+10−1
+10−2
+10−3
+10−4
+10−5
+errA
+O(10−2)
+O(10−5)
+O(10−7)
+O(10−9)
+O(10−12)
+Average �
+A−1 errors, for A ∼ 50 · N(0, 1) — CG
+ϵ
+10−3
+10−4
+10−5
+10−6
+10−7
+errN
+O(10−3)
+O(10−5)
+O(10−8)
+O(10−11) O(10−12)
+errrF
+O(10−3)
+O(10−5)
+O(10−7)
+O(10−10) O(10−12)
+We utilized Algorithm 1 in Newton’s method, for classifying images of four and nine from MNIST, by solving a regularized
+logistic regression minimization problem. For Algorithm 1, we used CG with a fixed number of iteration per column estimation.
+It is clear from Figure 4 that we require no more than 18 iterations per column estimate, for N = 785, to attain the optimal
+classification rate. With more than 18 CG iterations, the same classification rate was obtained.
+
+9
+14
+15
+16
+17
+18
+19
+CG iterarions per column
+0
+0.05
+0.1
+0.2
+0.3
+0.4
+0.45
+Classification error %
+Classification Error
+inversion with CG
+exact inversion
+Fig. 1. MNIST classification error, where Algorithm 1 is used in Newton’s method. In red, we depict the error when exact inversion was used.
+V. FEDERATED CODED MATRIX INVERSION
+In this section, we describe the proposed CMIM (also presented in [1]) which makes Algorithm 1 resilient to stragglers, and
+show how it can be applied to the FL scenario described in the introduction. The CMIM workflow is depicted in Figure 2.
+Our FL-scheme can be broken up in to four phases: (a) the coordinator shares elements β, H of a finite field with all the clients,
+(b) the clients each generate a pseudorandom permutation (PRP) σι, encrypt their corresponding data block Aι through a matrix
+polynomial fι(x), and broadcast {fι(x), σ−1
+ι
+} to the other clients, (c) the clients recover A, compute and encode their assigned
+task Wι, which is communicated to the coordinator, (d) the coordinator decodes once sufficiently many servers respond. It is also
+possible that β, H are determined collectively by the clients, or by a single client, which makes the data sharing secure against a
+curious and dishonest coordinator.
+Fig. 2. Algorithmic workflow of the CMIM, as proposed in [1]. The master shares f(x), an encoding analogous to (12), along with β, {η−1
+j
+}k
+j=1. The workers
+then recover A, compute their assigned tasks, and encode them according to G. Once k encodings Wι are sent back, �
+A−1 can be recovered.
+In our proposed FL approach, we assume there is a trustworthy coordinator who shares certain parameters to each of the k
+clients which constitute the network; e.g. hospitals in a health care network, each of which are comprised of multiple servers.
+What we present works for the case where the clients have local datasets of different sizes, {Ni}k
+i=1. This would result in the
+encoding functions fι(x) having different degrees, or their matrix coefficients being of a different size. In our setting we assume
+the servers are homogeneous, i.e. they have the same computational power. Therefore, equal computational loads are assigned to
+each of them. In order to keep the notation and size of the communication loads consistent, we assume w.l.o.g. that Aι ∈ RN×T
+for all ι ∈ Nk. If this is not the case, before fι(x) are determined, the clients could perform a data exchange phase (e.g. [13]), so
+that Ni = Nj for all i ̸= j. By this, it follows that the number of blocks does not have to be equal to the number of clients. The
+example we describe, is simply a motivation. A flowchart of our approach is presented in Figure 3.
+Moreover, in the case where M > N; for M = �k
+i=1 Ni, we can select a subset of features and/or samples, so that the resulting
+data matrix we consider is square. This can be interpreted as using the surrogate ˜A = SA, where S ∈ RN×M is an appropriate
+(sparse) sketching matrix for matrix inversion [32], which the workers agree on.
+First, in V-A we argue why all of A needs to be known by each of the workers, in order to recover entries or columns of its
+inverse. Then, in V-B we focus on phases (a) and (b), where we utilize Lagrange interpolation to securely share A among the
+workers. We discuss the computation tasks the workers are requested to compute, which are blocks of �
+A−1; and collectively
+correspond to the subroutine problems of Algorithm 1. In V-C we focus on (c) and (d), where we show how the servers encode
+their computations, and describe the coordinator’s decoding step. Optimality of BRS generator matrices in terms of the encoded
+communication loads is established in V-D.
+When assuming floating-point arithmetic, our approach introduces no numerical nor approximation errors. The errors are a
+consequence of using iterative solvers to estimate (4), which we utilize to linearly separate the computations. Therefore, if the
+workers can recover the optimal solutions to the underlying minimization problems, our scheme would be exact.
+
+>A-i =[Ai ... Ak
+W1
+X
+Wn
+X
+f(×)
+f(x)
+f(x)10
+Fig. 3. Flowchart of our proposal, where k = ni = 4 for all i ∈ N4.
+A. Knowledge of A is necessary
+A bottleneck when computing the inverse of a matrix; or estimating its columns, is that the entire matrix needs to be known.
+A single change in the matrix’s entries may result in a non-singular matrix, which conveys how sensitive Gaussian elimination is.
+Such problems are extensively studied in conditioning and stability of numerical analysis [30], and in perturbation theory. This is
+not a focus of our work.
+In the case where only one column is not known, one can determine the subspace in which the missing column lies, but without
+the knowledge of at least one entry of that column, it would be impossible to recover that column. Even with such an approach or
+a matrix completion algorithm, the entire A is determined before we proceed to inverting A; or performing linear regression to
+approximate Ab = ei as in (4).
+Another example, relating to our FL set up, is the case where one of the blocks is different. This could lead to drastic
+miscalculations. In the following example, we consider n = k = 2 and N = 4, where the second server sends a different
+block, which are indicated by a different color and font:
+A1 =
+�
+�
+�
+6
+2
+2
+-5
+0
+−1
+2
+0
+−5
+6
+-1
+-3
+5
+−3
+-4
+3
+�
+�
+�
+A2 =
+�
+�
+�
+6
+2
+−1
+−3
+0
+−1
+5
+6
+−5
+6
+3
+−2
+5
+−3
+1
+6
+�
+�
+� .
+It follows that ∥A−1
+1 ∥F ≈ 90.45, ∥A−1
+2 ∥F ≈ 1, and ∥A−1
+1
+− A−1
+2 ∥0 = 16; i.e. no entries of A−1
+1
+and A−1
+2
+are equal.
+Furthermore, by the data processing inequality [33, Corollary pg.35], the above imply that no less than N 2 information symbols
+can be known by each server, while hoping to approximate a column of A−1. Hence, all clients need full knowledge of each others
+information, and cannot communicate less than NT symbols to each other. This is a consequence of the fact that a dense vector is
+not recoverable from underdetermined linear measurements. They can however send an encoded version of their respective block
+Aι ∈ RN×T to the other clients consisting of NT symbols, determined by a modified Lagrange polynomial, which guarantees
+security against eavesdroppers.
+Similar cryptographic protocols date back to the SSS algorithm [34], which is also based on RS codes. This idea has extensively
+been exploited in LCC [5], yet differs from our approach.
+B. Phases (a), (b) — Data Encryption and Sharing
+Let k, γ ∈ Z+ be factors of N and T respectively, so that T = N
+k and Γ = T
+γ .3 The coordinator constructs a set of distinct
+interpolation points B = {βj}n
+j=1 ⊊ F×
+q , for q > n ⩾ γ.4 To construct this set, it suffices to sample β ∈ F×
+q ; any one of the
+φ(q − 1) primitive roots of Fq (φ is Euler’s totient function), which is a generator of the multiplicative group (F×
+q , ·), and define
+3If γ ∤ T, append 0T ×1 to the end of the first ˜γ = T(modγ) blocks which are each comprised of ˜Γ = ⌊ T
+γ ⌋ columns of Aι, while the remaining γ − ˜γ
+blocks are comprised of ˜Γ + 1 columns. Now, each block is of size T × (˜Γ + 1).
+4For the encodings of the Aι’s, γ points suffice, and we only need to require q > γ. We select B of cardinality n and require q > n ⩾ γ, in order to reuse B
+in our CCM.
+
+β,H
+β,H
+B,H
+β,H
+A
+A
+88
+fi(x),q1
+f2(x),02
+Clients share their corre
+sponding fi(×) and o,-1
+Clients recover A = Ai A2 A3 A4
+Servers carry out computations, and each
+client sends back W, to the coordinator
+=
+Once the threshold is met,11
+each point as βj = βj. Then, a random multiset H = {ηj}γ
+j=1 ∈ 2F×
+q of size γ is generated, i.e. repetitions in H are allowed,
+which will be used to remove the structure of the Lagrange coefficients, as the adversaries could exploit their structure to reveal
+β.
+The element β and set H, are broadcasted securely to all the workers through a public-key cryptosystem, e.g. RSA or McEliece.
+Matrices Aι are partitioned into γ blocks
+Aι =
+�
+A1
+ι · · · Aγ
+ι
+�
+where Ai
+ι ∈ RN×Γ, ∀i ∈ Nγ,
+(11)
+and each client generates a PRP σι ∈ Sγ. The blocks {Aι}k
+ι=1 are encrypted locally through the univariate polynomials
+fι(x) =
+γ
+�
+j=1
+Aj
+ι · ησι(j)
+�
+��
+l̸=j
+x − βl
+βj − βl
+�
+�
+(12)
+for which fι(βj) = ησι(j)Aj
+ι.
+The clients then broadcast {fι(x), σ−1
+ι
+} to each other, and their servers can then recover all Aι’s as follows:
+Aι =
+�
+ησ−1
+ι
+(1)f(β1) · · · ησ−1
+ι
+(γ)f(βγ)
+�
+∈ RN×T .
+(13)
+The coefficients of fι(x) are comprised of NΓ symbols, thus, each polynomial consists of a total of NT symbols, which is the
+minimum number of symbols needed to be communicated. The PRP σι is generated locally by the clients, to ensure that each
+fι(x) differs by more than just the matrix partitions.
+We assume Kerckhoffs’ principle, which states that everyone has knowledge of the system, including the messages fι(x). For
+the proposed CMIM, as long as {β, H} and σ−1
+ι
+are securely communicated, even if fι(x) is revealed, the block Aι is secure
+against polynomial-bounded adversaries (this is the security level assumed by the cryptosystems used for the communication).
+Proposition 5. The encryptions of Aι through fι(x), are as secure against eavesdroppers as the public-key cryptosystems which
+are used when broadcasting {β, H} and σ−1
+ι
+. To recover Aι, an adversary needs to intercept both communications, and break
+both cryptosystems.
+Proof. We prove this by contradiction. Assume that an adversary was able to reverse the encoding fι(x) of Aι. This implies that
+he was able to reveal β and σι(H) := {ησι(j)}γ
+j=1. The only way to reveal these elements, is he was able to both intercept and
+decipher the public-key cryptosystem used by the coordinator, which contradicts the security of the cryptosystem.
+In order to invert the multiplications of σι(H) for each of the evaluations of fι(x), both H and σ−1
+ι
+need to be known. To do so,
+the adversary needs to intercept both the communication between the coordinator and the clients, and the communication between
+the clients, as well as breaking both the cryptosystems used to securely carry out these communications.
+■
+C. Phases (c), (d) — Computations, Encoding and Decoding
+At this stage, the workers have knowledge of everything they need in order to recover A, before they carry out their computation
+tasks. By (13), the recovery is straightforward.
+For Algorithm 1, any CCM in which the workers compute an encoding of partitions of the resulting computation E =
+�
+E1 · · · Ek
+�
+could be utilized. It is crucial that the encoding takes place on the computed tasks {Ei}k
+i=1 in the scheme, and
+not the assigned data or partitions of the matrices that are being computed over (such CMM leverage the linearity of matrix
+multiplication), otherwise the algorithm could potentially not return the correct approximation. This also means that utilizing such
+encryption approaches (e.g. [5]) for guaranteeing security against the workers, is not an option. We face these restrictions due to
+the fact that matrix inversion is a non-linear operator.
+The computation tasks Ei correspond to a partitioning �
+A−1 =
+� ˆ
+A1 · · ·
+ˆ
+Ak
+�
+, of our approximation from Algorithm 1. We
+propose a linear encoding of the computed blocks { ˆ
+Ai}k
+i=1 based on generators satisfying (2). Along with the proposed decoding
+step, we have a MDS-based CCM for matrix inversion.
+We consider the same parameters as in V-B, in order to reuse B in the proposed CMIM. Each ˆ
+Ai is comprised of T distinct but
+consecutive approximations of (4), i.e.
+ˆ
+Ai =
+�ˆb(i−1)T +1 · · · ˆbiT
+�
+∈ RN×T
+∀i ∈ Nk,
+which could also be approximated by iteratively solving
+ˆ
+Ai ≈ arg min
+B∈RN×T
+���AB −
+�
+e(i−1)T +1 · · · eiT
+���2
+F
+�
+.
+Without loss of generality, we assume that the workers use the same algorithms and parameters for estimating the columns
+{ˆbi}N
+i=1. Therefore, workers allocated the same tasks are expected to get equal approximations in the same amount of time.
+
+12
+For our CCM, we leverage BRS generator matrices for both the encoding and decoding steps. We adapt the GC framework,
+so we need an analogous condition to a⊤
+I G = ⃗1 for the CMIM; in order to invoke Algorithm 1. The condition we require is
+˜DI ˜G = IN, for an encoding-decoding pair ( ˜G, ˜DI).
+From our discussion on BRS codes in III-A, we set ˜G = IT ⊗ G and ˜DI = IT ⊗ (GI)−1 for any given set of k responsive
+servers indexed by I. The index set of blocks requested from the ιth worker to compute is denoted by Jι, and has cardinality w.
+The workers’ encoding steps correspond to
+˜G · ( �
+A−1)⊤ = (IT ⊗ G) ·
+�
+��
+ˆ
+A⊤
+1
+...
+ˆ
+A⊤
+k
+�
+�� =
+�
+�
+�
+�
+�
+�
+�
+j∈J1
+pj(β1) · ˆ
+A⊤
+j
+...
+�
+j∈Jn
+pj(βn) · ˆ
+A⊤
+j
+�
+�
+�
+�
+�
+�
+(14)
+which are carried out locally, once they have computed their assigned tasks. We denote the encoding of the ιth worker by Wι ∈
+CT ×N, i.e. Wι = �
+j∈Jι pj(βι) · ˆ
+A⊤
+j , which are sent to the coordinator. The received encoded computations by any distinct k
+servers indexed by I, constitute ˜GI · ( �
+A−1)⊤.
+Lemma 1 implies that as long as k workers respond, the approximation �
+A−1 is recoverable. Moreover, the decoding step
+reduces to a matrix multiplication of k × k matrices. Applying HI to a square matrix can be done in O(k2 log k), through the
+FFT algorithm. The prevailing computation in our decoding, is applying P−1. The decoding step is
+˜DI ·
+�
+˜GI · ( �
+A−1)⊤�
+=
+�
+IT ⊗ (GI)−1�
+·
+�
+IT ⊗ GI
+�
+· ( �
+A−1)⊤
+= (IT · IT ) ⊗
+�
+(GI)−1 · GI
+�
+· ( �
+A−1)⊤
+= IT ⊗ Ik · ( �
+A−1)⊤
+= ( �
+A−1)⊤
+and our scheme is valid.
+The above CCM therefore has a linear encoding done locally by the servers (14), is MDS since s = d − 1, and its decoding
+step reduces to computing and applying G−1
+I
+(Lemma 1). The security of the encodings rely on the secrecy of B, which were
+sent from the coordinator to the workers. For an additional security layer, the interpolation points of B could instead be defined as
+βj = βπ(j), for π ∈ Sn a PRP. In this case, π−1 would also need to be securely broadcasted.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+106
+0
+20
+40
+60
+80
+100
+120
+4.5
+5
+5.5
+6
+6.5
+7
+0
+20
+40
+60
+80
+100
+120
+Fig. 4. Comparison of decoding complexity, when naive matrix inversion is used (so O(k3)) compared to the decoding step implied by Lemma 1, for n = 200
+and varying s. We also provide a logarithmic scale comparison.
+With the above framework, any sparsest-balanced generator MDS matrix [23] would suffice, as long as it satisfies the MDS
+theorem [35]. By Lemma 1, if we set k = Ω(
+√
+N) (similar to [7]), the decoding step could then be done in O(N ω/2) = o(N 1.187),
+which is close to linear in terms of N.
+Theorem 6. Let G ∈ Fn×k be a generator matrix of any MDS code over F, for which ∥G(j)∥0 = n − k + 1 and ∥G(i)∥0 = w
+for all (i, j) ∈ Nn × Nk. By utilizing Algorithm 1, we can devise a linear MDS coded matrix inversion scheme; through the
+encoding-decoding pair ( ˜G, ˜DI).
+Proof. The encoding coefficients applied locally by each of the n workers correspond to a row of G. The encodings of all the
+workers then correspond to ˜G · ( �
+A−1)⊤, for ˜G = IT ⊗ G, as in (14). Consider any set of responsive workers I of size k, whose
+encodings constitute ˜GI ·( �
+A−1)⊤. By the MDS theorem, GI is invertible. Hence, the decoding step reduces to inverting GI; i.e.
+˜DI = IT ⊗ (GI)−1, and is performed online.
+■
+Constructions based on cyclic MDS codes, which have been used to devise GC schemes [36], can also be considered. These
+encoding matrices are not sparsest-balanced, which makes them suitable when considering heterogeneous workers.
+Proposition 7. Any cyclic [n, k] MDS code C over F ∈ {R, C} can be used to devise a coded matrix inversion encoding-decoding
+pair ( ˜G, ˜DI).
+
+13
+Proof. Consider a cyclic [n, n − s] MDS code C over F ∈ {R, C}. Recall that from our assumptions, we have s = n − k. By [36,
+Lemma 8], there exists a codeword g1 ∈ C of support d = s + 1, i.e. ∥g1∥0 = d. Since C is cyclic, it follows that the cyclic shifts
+of g1 also lie in C. Denote the n − 1 consecutive cyclic shifts of g1 by {gi}n
+i=2 ⊊ C ⊊ F1×n, which are all distinct. Define the
+cyclic matrix
+¯G :=
+�
+�
+|
+|
+|
+g⊤
+1
+g⊤
+2
+. . .
+g⊤
+n
+|
+|
+|
+�
+� ∈ Fn×n.
+Since ∥gi∥0 = d and gi is a cyclic shift of gi−1 for all i > 1, it follows that ∥ ¯G(i)∥0 = ∥ ¯G(j)∥0 = d for all i, j ∈ Nn, i.e. ¯G
+is sparsest and balanced. If we erase any s = n − k columns of ¯G, we get G ∈ Fn×k. By erasing arbitrary columns of ¯G, the
+resulting G is not balanced, i.e. we have ∥G(i)∥0 ̸= ∥G(j)∥0 for some pairs i, j ∈ Nn. Similar to our construction based on BRS
+generator matrices, we define the encoding matrix to be ˜G = IT ⊗ G. The local encodings are then analogous to (14).
+Consider an arbitrary set of k non-straggling workers I ⊊ Nn, and the corresponding matrix GI ∈ Fk×k. By [36, Lemma 12,
+B4.], GI is invertible. The decoding matrix is then ˜DI = IT ⊗ (GI)−1, and the condition ˜DI ˜G = IN is met.
+■
+D. Optimality of MDS BRS Codes
+Under the assumption that k = n − s, by utilizing the BRSq[n, k] generator matrices, we achieved the minimum possible
+communication load from the workers to the coordinator. From our discussion in V-A, we cannot hope to receive an encoding
+of less than N 2/k symbols; when we require that k workers respond with the same amount of information symbols in order
+to recover �
+A−1 ∈ RN×N, unless we make further assumptions on the structure of A and A−1. Each encoding Wι consists
+of NT = N 2/k symbols, so we have achieved the lower bound on the minimum amount of information needed to be sent to
+the coordinator. Hence, Wι ∈ CT ×N for any sparsest-balance generator MDS matrix. This also holds true for other generator
+matrices which can be used in Theorem 6, as the encodings are linear (e.g. Proposition 7).
+We also require the workers to estimate the least possible number of columns for the given recovery threshold k. For our choice
+of parameters, the bound of [20, Theorem 1] is met with equality. That is, for all i ∈ Nn:
+∥G(i)∥0 = w = k
+n · d = k
+n · (n − k + 1) ,
+which means that for homogeneous workers, we cannot get a sparser generator matrix. This, along with the requirement that GI
+should be invertible for all possible I, are what we considered in (2).
+VI. CONCLUSION AND FUTURE WORK
+In this paper, we addressed the problem of approximate computation of the inverse of a matrix distributively in a FL setting,
+under the possible presence of straggling workers. We provided approximation error bounds for our approach, as well as security
+and recovery guarantees. We also provided numerical experiments that validated our proposed approach.
+There are several interesting future directions. One is looking into the issue of numerical stability of the BRS approach, and
+exploring other suitable generator matrices, e.g. circulant permutation and rotation matrices [37]. Another direction, is leveraging
+approximate CCMs. The techniques of [22], [38] suggest that carefully selecting interpolation points may lead to more efficient
+(approximate) schemes. In terms of coding-theory, it would be interesting to see if it is possible to reduce the complexity of our
+decoding step. Specifically, could well-known RS decoding algorithms such as the Berlekamp-Welch algorithm be exploited?
+Another important extension is to reduce the communication rounds when computing the pseudoinverse through our approach.
+This depends on the CMM which is being utilized, though using different ones for each of the two multiplications may also be
+beneficial.
+Tribute to Alex Vardy: As this is a special issue dedicated to the memory Alexander Vardy, we mention how this paper relates
+to some of his work. Even though Alex had not worked on CC, his contributions to RS codes are immense. A focus of this paper
+is to reduce the decoding complexity of the proposed BRS-based CCM, while in [39] it was shown that ML decoding of RS
+codes is NP-hard. Another highly innovative work of Vardy’s is [40], in which the ‘Parvaresh-Vardy codes’ were introduced;
+and the associated list-decoding algorithm was shown to yield an improvement over the Guruswami–Sudan algorithm. This was
+subsequently improved by Guruswami and Rudra [41], whose techniques were exploited in [42] to introduce list-decoding in CC.
+APPENDIX A
+ADDITIONAL MATERIAL AND BACKGROUND
+In this appendix, we include material and background which was used in our derivations. First, we recall what an ϵ-optimal
+solution/point is, which was used in the proof of Proposition 4. Next, we state the MDS Theorem and the BCH Bound. We
+then give a brief overview of the GC scheme from [8], to show how it differs from our coded matrix inversion scheme. We also
+explicitly give their construction of a balanced mask matrix M ∈ {0, 1}n×k, which we use for the construction of the BRS
+generator matrices. Lastly, we illustrate a simple example of the encoding matrix.
+
+14
+Definition 8 ( [43]). A point ¯x is said to be an ϵ-optimal solution/point to a minimization problem with objective function f(x),
+if for any x, it holds that f(x) ⩾ f(¯x) − ϵ, where ϵ ⩾ 0. When ϵ = 0, an ϵ-optimal solution is an exact minimizer.
+Theorem 9 (MDS Theorem — [35]). Let C be a linear [n, k, d] code over Fq, with G, H the generator and parity-check matrices.
+Then, the following are equivalent:
+1) C is a MDS code, i.e. d = n − k + 1
+2) every set of n − k columns of H is linearly independent
+3) every set of k columns of G is linearly independent
+4) C⊥ is a MDS code.
+Theorem 10 (BCH Bound — [15], [25]). Let p(x) ∈ Fq[x]\{0} with t cyclically consecutive roots, i.e. p(αj+ι) = 0 for all
+ι ∈ Nt. Then, at least t + 1 coefficients of p(x) are nonzero.
+Algorithm 2: MaskMatrix(n, k, d) [8]
+Input: n, k, d ∈ Z+ s.t. n > d, k and w = kd
+n
+Output: row-balanced mask matrix M ∈ {0, 1}n×k
+M ← 0n×k
+for j = 0 to k − 1 do
+for i = 0 to d − 1 do
+ι ← (i + jd + 1) mod n
+Mr,ι ← 1
+end
+end
+return M
+Even though this was not pointed out in [8], Algorithm 2 does not always produce a mask matrix of the given parameters when
+we select d < n/2. This is why in our work we require d ⩾ n/2.
+The decomposition G = HP is utilized in the GC scheme of [8]. Each column of G corresponds to a partition of the data
+whose partial gradient is to be computed. The polynomials are judiciously constructed in this scheme, such that the constant term
+of each polynomial is 1 for all polynomials, thus P(1) = ⃗1. By this, the decoding vector a⊤
+I is the first row of G−1
+I , for which
+a⊤
+I GI = e⊤
+1 . A direct consequence of this is that a⊤
+I BI = e⊤
+1 T = T(1) = ⃗1, which is the objective for constructing a GC
+scheme.
+A. Generator Matrix Example
+As an example, consider the case where n = 9, k = 6 and d = 6, thus w = kd
+n = 4. Then, Algorithm 2 produces
+M =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∈ {0, 1}9×6 .
+For our CCM, this means that the ith worker computes the blocks indexed by supp(M(i)), e.g. supp(M(1)) = {1, 2, 4, 5}. We
+denote the indices of the respective task allocations by Ji = supp(M(i)). The entries of the generator matrix G are the evaluations
+of the constructed polynomials (3) at each of the evaluation points B = {βi}n
+i=1, i.e. Gij = pj(αi). This results in:
+G =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+p1(β1)
+p2(β1)
+0
+p4(β1)
+p5(β1)
+0
+p1(β2)
+p2(β2)
+0
+p4(β2)
+p5(β2)
+0
+p1(β3)
+p2(β3)
+0
+p4(β3)
+p5(β3)
+0
+p1(β4)
+0
+p3(β4)
+p4(β4)
+0
+p6(β4)
+p1(β5)
+0
+p3(β5)
+p4(β5)
+0
+p6(β5)
+p1(β6)
+0
+p3(β6)
+p4(β6)
+0
+p6(β6)
+0
+p2(β7)
+p3(β7)
+0
+p5(β7)
+p6(β7)
+0
+p2(β8)
+p3(β8)
+0
+p5(β8)
+p6(β8)
+0
+p2(β9)
+p3(β9)
+0
+p5(β9)
+p6(β9)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+
+15
+APPENDIX B
+DISTRIBUTED PSEUDOINVERSE
+For full-rank rectangular matrices A ∈ RN×M where N > M, one resorts to the left Moore–Penrose pseudoinverse A† ∈
+RM×N, for which A†A = IM. In Algorithm 3, we present how to approximate the left pseudoinverse of A, by using the fact that
+A† = (A⊤A)−1A��; since A⊤A ∈ GLN(R). The right pseudoinverse A† = A⊤(AA⊤)−1 of A ∈ RM×N where M < N,
+can be obtained by a modification of Algorithm 3.
+Just like the inverse, the pseudoinverse of a matrix also appears in a variety of applications. Computing the pseudoinverse of
+A ∈ RN×M for N > M is even more cumbersome, as it requires inverting the Gram matrix A⊤A. For this subsection, we
+consider a full-rank matrix A.
+One could naively attempt to modify Algorithm 1 in order to retrieve A† such that A†A = IM, by approximating the rows
+of A†. This would not work, as the underlying optimization problems would not be strictly convex. Instead, we use Algorithm
+3 to estimate the rows of B−1 := (A⊤A)−1, and then multiply the estimate �
+B−1 by A⊤. This gives us the approximation
+�
+A† = �
+B−1 · A⊤.
+The drawback of Algorithm 3 is that it requires two additional matrix multiplications, A⊤A and �
+B−1A⊤. We overcome this
+barrier by using a CMM scheme twice, to recover �
+A† in a two or three-round communication CC approach. These are discussed
+in below.
+Bounds on errF ( �
+A−1) and errrF ( �
+A−1) can be established for both algorithms, specific to the black-box least squares algorithm
+being utilized. This is left for future work.
+Algorithm 3: Estimating A†
+Input: full-rank A ∈ RN×M where N > M
+B ← A⊤A
+for i=1 to M do
+ˆci = arg minc∈R1×M
+�
+gi(c) := ∥cB − e⊤
+i ∥2
+2
+�
+ˆbi ← ˆci · A⊤
+end
+return �
+A† ←
+�
+ˆb⊤
+1 · · · ˆb⊤
+M
+�⊤
+▷ �
+A†(i) = ˆbi
+Corollary 11. For full-rank A ∈ RN×M with N > M, we have errF (�
+A†) ⩽
+√
+Mϵ·κ2
+√
+2σmin(A)3 and errrF (�
+A†) ⩽
+√
+Mϵ·κ2
+√
+2σmin(A)2 when
+using SD to solve the subroutine optimization problems of Algorithm 3, with termination criteria ∥∇gi(c[t])∥2 ⩽ ϵ.
+Proof. From (10), it follows that
+∥B−1ei − ˆc⊤
+i ∥2 ⩽
+ϵ/
+√
+2
+σmin(B)2 =
+ϵ/
+√
+2
+σmin(A)4 =: δ .
+The above bound implies that for each summand of the Frobenius error; ∥ˆbi − A†
+(i)∥2 = ∥ˆciA⊤ − e⊤
+i · B−1A⊤∥2, we have
+∥ˆbi − A†
+(i)∥2 ⩽ δ∥A⊤∥2. Summing the right hand side M times, we get that
+errF (�
+A†)2 ⩽ M · (δ∥A⊤∥2)2
+= Mϵ2 · σmax(A)2
+σmin(A)8
+= Mϵ2 · κ2
+2
+σmin(A)6 .
+By taking the square root, we have shown the first claim.
+Since 1/σmin(A) = ∥A†∥2 ⩽ ∥A†∥F , it then follows that
+errrF (�
+A†) = errF (�
+A†)
+∥A†∥F
+⩽ errF (�
+A†)
+∥A†∥2
+=⩽
+√
+Mϵ · κ2
+√
+2σmin(A)2 ,
+which completes the proof.
+■
+A. Pseudoinverse from Polynomial CMM
+One approach to leverage Algorithm 3 in a two-round communication scheme is to first compute B = A⊤A through a CMM
+scheme, then share B with all the workers who estimate the rows of �
+B−1, and finally use another CMM to locally encode the
+
+16
+estimated columns with blocks of A⊤; to recover �
+A† = �
+B−1 · A⊤. Even though there are only two rounds of communication, the
+fact that we have a local encoding by the workers results in a higher communication load overall. An alternative approach which
+circumvents this issue, uses three-rounds of communication.
+For this approach, we use the polynomial CMM scheme from [7] twice, along with our coded matrix inversion scheme. This
+CMM has a reduced communication load, and minimal computation is required by the workers. To have a consistent recovery
+threshold across our communication rounds, we partition A as in (11) into ¯k = √n − s =
+√
+k blocks. Each block is of size
+N × ¯T, for ¯T = M
+k . The encodings from [7] of the partitions {Aj}¯k
+j=1 for carefully selected parameters a, b ∈ Z+ and distinct
+elements γi ∈ Fq, are
+˜Aa
+i =
+k
+�
+j=1
+Ajγ(j−1)a
+i
+and
+˜Ab
+i =
+k
+�
+j=1
+Ajγ(j−1)b
+i
+for each worker indexed by i. Thus, each encoding is comprised of N ¯T symbols. The workers compute the product of their
+respective encodings ( ˜Aa
+i )⊤ · ˜Ab
+i. The decoding step corresponds to an interpolation step, which is achievable when ¯k2 = k many
+workers respond5, which is the optimal recovery threshold for CMM. Any fast polynomial interpolation or RS decoding algorithm
+can be used for this step, to recover B.
+Next, the master shares B with all the workers (from V-A, this is necessary), who are requested to estimate the column-blocks
+of �
+B−1
+�
+B−1 =
+�
+¯B1 · · · ¯Bk
+�
+where ¯Bj ∈ RM× ¯T ∀j ∈ Nk
+(15)
+according to Algorithm 1. We can then recover �
+B−1 by our BRS based scheme, once k workers send their encoding.
+For the final round, we encode �
+B−1 as
+˜Ba
+i =
+k
+�
+j=1
+¯Bjγ(j−1)a
+i
+which are sent to the respective workers. The workers already have in their possession the encodings ˜Ab
+i. We then carry out the
+polynomial CMM where each worker is requested to send back ( ˜Ba
+i )⊤ · ˜Ab
+i. The master server can then recover �
+A†.
+Theorem 12. Consider G ∈ Fn×k as in Theorem 6. By using any CMM, we can devise a matrix pseudoinverse CCM by utilizing
+Algorithm 3, in two-rounds of communication. By using polynomial CMM [7], we achieve this with a reduced communication
+load and minimal computation, in three-rounds of communication.
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+

9NE2T4oBgHgl3EQflwcz/content/tmp_files/2301.03991v1.pdf.txt

@@ -0,0 +1,1075 @@
+Constraining cosmological parameters from
+N-body simulations with Variational Bayesian
+Neural Networks
+H´ector J. Hort´ua 1,2,∗, Luz ´Angela Garc´ıa 3 and Leonardo Casta˜neda C. 4
+1 Grupo Signos, Departamento de Matem´aticas, Universidad el Bosque, Bogot´a,
+Colombia.
+2 Maestr´ıa en Ciencia de Datos, Universidad Escuela Colombiana de Ingenier´ıa
+Julio Garavito Bogot´a, Colombia.
+3 Universidad ECCI. Cra. 19 No. 49-20, Bogot´a, Colombia, C´odigo Postal 111311.
+4Observatorio Astron´omico Nacional, Universidad Nacional de Colombia, Bogot´a,
+Colombia.
+Correspondence*:
+-
+[email protected]
+ABSTRACT
+Methods based on Deep Learning have recently applied on astrophysical parameter recovery
+thanks to their ability to capture information from complex data. One of these methods are the
+approximate Bayesian Neural Networks (BNNs) which have demonstrated to yield consistent
+posterior distribution into the parameter space, helpful for uncertainty quantification. However,
+as any modern neural networks, they tend to produce overly confident uncertainty estimates,
+and can introduce bias when BNNs are applied to data. In this work, we implement multiplicative
+normalizing flows (MNFs), a family of approximate posteriors for the parameters of BNNs with the
+purpose of enhancing the flexibility of the variational posterior distribution, to extract Ωm, h, and
+σ8 from the QUIJOTE simulations. We have compared this method with respect to the standard
+BNNs, and the flipout estimator. We found that MNFs combined with BNNs outperform the other
+models obtaining predictive performance with almost one order of magnitude larger that standard
+BNNs, σ8 extracted with high accuracy (r2 = 0.99), and precise uncertainty estimates. The
+latter implies that MNFs provide more realistic predictive distribution closer to the true posterior
+mitigating the bias introduced by the variational approximation, and allowing to work with well
+calibrated networks.
+Keywords: cosmology, N-body simulations, parameter estimation, artificial intelligence, deep neural networks
+1
+INTRODUCTION
+Cosmological simulations offer one of the most powerful ways to understand the initial conditions of
+the Universe and improve our knowledge on fundamental physics (1). They open also the possibility to
+fully explore the growth of structure in both the linear and non-linear regime. Currently, the concordance
+cosmological model, Λ-CDM, gives an accurate description of most of the observations from early to late
+stages of the Universe using a set of few parameters (2). Recent observations from Cosmic Microwave
+Background (CMB) have provided such accurate estimation for the cosmological parameters, and prompted
+a tension with respect to local scales measurement, along with a well-known degeneracy on the total
+non-relativistic matter density parameters (3, 4, 5). Conventionally, the way to capture information from
+1
+arXiv:2301.03991v1  [astro-ph.IM]  9 Jan 2023
+
+Hort´ua et al.
+Parameter estimation via BNNs
+astronomical observations is to compare summary statistics from data against theory predictions. However,
+two major difficulties arise: First, it is not well understood what kind of estimator, or at which degree of
+approximation of order statistic should be better to extract the maximum information from observations. In
+fact, the most common choice is the power spectrum(PS) which has shown to be a powerful tool for making
+inference (2). However, It is well known that PS is not able to fully characterize the statistical properties
+of non-Gaussian density fields, yielding that it would not be suitable for upcoming Large Scale Structure
+(LSS) or 21-cm signals which are highly non-Gaussian (6, 7, 8). Then, PS will miss relevant information if
+only this statistic is used for parameter recovery (9). Second, Cosmologists will require to store and process
+a large number of data, which can be very expensive. Clearly, sophisticated computational tools along with
+new perspectives on data collection, storage, and analysis must be developed in order to interpret these
+observations (10).
+In recent years, artificial intelligence (AI), and Deep Neural Networks (DNNs) have emerged as promising
+tools to tackle the aforementioned difficulties in the cosmological context due to its capability for
+learning relationships between variables in complex data, outperform traditional estimators, and handle
+the demanding computational needs in Astrophysics and Cosmology (10). These standard DNNs have
+been used on a variety of tasks because of their potential for solving inverse problems. However, they
+are prone to overfitting due to the excessive number of parameters to be adjusted, and the lack of
+explanations of their predictions for given instances (11). The latter is crucial for cosmological analysis
+where assessing robustness and reliability of the model predictions are imperative. This problem can be
+addressed by endowing DNNs with probabilistic properties that permit quantifying posterior distributions
+on their outcomes, and provide them with predictive uncertainties. One of these approaches is the
+use of Bayesian Neural Networks (BNNs) comprised of probabilistic layers that capture uncertainty
+over the network parameters (weights), and trained using Bayesian inference (12). Several works have
+utilized BNNs in cosmological scenarios where the combination of DNNs (through Convolutional Neural
+Networks, CNNs) and probabilistic properties, allow to build models adapted to non-Gaussian data
+without requiring a priori choice summary statistic (9, 13, 14, 15), along with quantifying predictive
+uncertainties (16, 17, 18, 19, 20, 21). Indeed, BNNs permit to infer posterior distributions instead of point
+estimates for the weights. These distributions capture the parameter uncertainty, and by subsequently
+integrating over them, we acquire uncertainties related to the network outputs. Nevertheless, obtaining the
+posterior distributions is an intractable task, and approximate techniques such as a Variational Inference(VI)
+must be used in order to put them into practice (22). Despite the approximate posterior distribution over
+the weights employed in VI clearly providing fast computations for inference tasks, they can also introduce
+a degree of bias depending on how complex(or simple) the choice of the approximate distribution family
+is (23). This issue yields overconfident uncertainty predictions and an unsatisfactory closeness measurement
+with respect to the true posterior. In (17, 18), the authors included normalizing flows on the top of BNNs to
+give the joint parameter distribution more flexibility. However, that approach is not implemented into the
+Bayesian framework, preserving the bias.
+In this paper, we attempt to enhance the flexibility of the approximate posterior distribution over the
+weights of the network by employing multiplicative normalizing flows, resulting in accurate and precise
+uncertainty estimates provided by BNNs. We apply this approach to N-body simulations taken from
+QUIJOTE dataset (24) in order to show how BNNs can take not only advantage of non-Gaussian signals
+without requiring a specifying the summary statistic (such as PS) but also, increase the posterior complexity,
+as they yield much larger performance improvements. This paper is organized as follows. Section 2 offers a
+summary of the BNNs framework and a detailed description of Normalizing flow implementation. Section 3
+describes the dataset and analysis tools used in this paper. Numerical implementation and configuration for
+2
+
+Hort´ua et al.
+Parameter estimation via BNNs
+BNNs are described in Section 4. Section 5 presents the results we obtained by training BNNs taking into
+account different approaches and we display the inference of cosmological parameters. It also outlines the
+calibration diagrams to determine the accuracy of the uncertainty estimates. Finally, Section 6 draws the
+main conclusions of this work and possible further directions to the use of BNNs in Cosmology.
+2
+VARIATIONAL BAYESIAN NEURAL NETWORKS
+Here we go into detail about Bayesian Neural Networks (BNNs), and their implementation to perform
+parameter inference. We start with a brief introduction, before focusing on improving the variational
+approximation. We remind the reader to refer to (25, 26, 22) for further details.
+2.1
+Approximate BNNs
+The goal of BNNs is to infer the posterior distribution p(w|D) over the weights w of the network after
+observing the data D = (X, Y ). This posterior can be obtained from Bayes law: p(w|D) ∼ p(D|w)p(w),
+given a likelihood function p(D|w), and a prior on the weights p(w). Once the posterior has been computed,
+the probability distribution on a new test example x∗ is given by
+p(y∗|x∗, D) =
+�
+w
+p(y∗|x∗, w)p(w|D)dw,
+(1)
+where p(y∗|x∗, w) is the predictive distribution for a given value of the weights. For neural networks,
+however, computing the exact posterior is intractable, so one must resort to approximate BNNs for
+inference (26). A popular method to approximate the posterior is variational inference(VI) (22). Let
+q(w|θ) be a family of simple distributions parameterized by θ. So, the goal of VI is to select a distribution
+q(w|θ∗) such that θ∗ minimizes KL
+�
+q(w|θ)
+��p(w|D)
+�
+, being KL[·∥·] the Kullback-Leibler divergence.
+This minimization is equivalent to maximizing the evidence lower bound (ELBO) (26)
+ELBO(θ) = Eq(w|θ)
+�
+log p(Y |X, w)
+�
+− KL
+�
+q(w|θ)
+��p(w)
+�
+,
+(2)
+where Eq(w|θ)[log p(Y |X, w)] is the expected log-likelihood with respect to the variational posterior and
+KL[q(w|θ)||p(w)] is the divergence of the variational posterior from the prior. We can observe from Eq. 2
+that the KL divergence acts as a regularizer that encourages the variational posterior moves towards
+the modes of the prior. A common choice for the variational posterior is a product of independent (i.e.,
+mean-field) Gaussian distributions, one distribution for each parameter w in the network (25)
+q(w|θ) =
+�
+ij
+N(w; µij, σ2
+ij)
+(3)
+being i and j the indices of the neurons from the previous layer and the current layer respectively. Applying
+the reparametrization trick we arrive at wij = µij + σij ∗ ϵij, where ϵij is drawn from a standard normal
+distribution. Furthermore, if the prior is also a product of independent Gaussians, the KL divergence
+between the prior and the variational posterior be computed analytically, which makes this approach
+computationally efficient.
+2.1.1
+Flipout
+In case where sampling from q(w|θ) is not fully independently for different examples in a mini-batch, we
+well obtain gradient estimates with high variance. Flipout method provides an alternative to decorrelate the
+3
+
+Hort´ua et al.
+Parameter estimation via BNNs
+gradients within a mini batch by implicitly sampling pseudo-independent weights for each example (27).
+The method requires two assumptions about the properties of q(w|θ): symmetric with respect to zero,
+and the weights of the network are independent. Under these assumptions, the distribution is invariant to
+element wise multiplication by a random sign matrix ˆr, i.e., ˆw = w◦ˆr, implies that w ∼ q(w) ≈ ˆw ∼ ˆq( ˆw).
+Therefore, the marginal distribution over gradients computed for individual examples will be identical to
+the distribution computed using shared weights samples. Hence, Flipout achieves much lower variance
+updates when averaging over a mini batch. We validate this approach experimentally by comparing against
+Multiplicative normalizing flows.
+2.2
+Uncertainty in BNNs
+BNNs offer a groundwork to incorporate from the posterior distribution both, the uncertainty inherent to
+the data (aleatoric uncertainty), and the uncertainty in the model parameters due to a limited amount of
+training data (epistemic uncertainty) (28). Following (16), assuming that the top of the BNNs consist of a
+mean vector µ ∈ RN and a covariance matrix Σ ∈ RN(N+1)/21, and for a given fixed input x∗, T forward
+passes of the network are computed, obtaining for each of their mean µt and covariance matrix Σt. Then,
+an estimator for approximate the predictive covariance can be written as
+�
+Cov(y∗, y∗|x∗) ≈ 1
+T
+T
+�
+t=1
+Σt
+�
+��
+�
+Aleatoric
++ 1
+T
+T
+�
+t=1
+(µt − µ)(µt − µ)T
+�
+��
+�
+Epistemic
+,
+(4)
+with µ = 1
+T
+�T
+t=1 µt. Notice that in case Σ is diagonal, and σ2 = diag(Σ), the last equation reduces to the
+results obtained in (29, 30)
+�
+Var(y∗|x∗) ≈ 1
+T
+T
+�
+t=1
+σ2
+t
+�
+��
+�
+Aleatoric
++ 1
+T
+T
+�
+t=1
+(µt − ¯µ)2
+�
+��
+�
+Epistemic
+.
+(5)
+In this scenario, BNNs can be used to learn the correlations between the the targets and produce estimates
+of their uncertainties. Unfortunately, the uncertainty computed from Eqs. 4, 5, tends to be miscalibrated, i.e.,
+the predicted uncertainty (taking into account both epistemic and aleatoric uncertainty) is underestimated
+and does not allow robust detection of uncertain predictions at inference. Therefore, calibration diagrams
+along with methods to jointly calibrate aleatoric and epistemic uncertainties, must be employed before
+inferring predictions from BNNs (31). We come back to this point in Section 5.
+2.3
+Multiplicative normalizing flows
+As mentioned previously, the most common family for the variational posterior used in BNNs is the
+mean-field Gaussian distributions defined in Eq. 3. This simple distribution is unable to capture the
+complexity of the true posterior. Therefore, we expect that increasing the complexity of the variational
+posterior, BNNs achieve significant performance gains since we are now able to sample from a complicate
+distribution that more closely resembles the true posterior. Certainly, transforming the variational posterior
+must be followed with fast computations and still being numerically tractable. We now describe in detail
+the Multiplicative Normalizing Flows (MNFs) method that provides flexible posterior distributions in an
+1 Where the targets y ∈ RN.
+4
+
+Hort´ua et al.
+Parameter estimation via BNNs
+efficient way by employing auxiliary random variables and normalizing flows proposed by (32). MNFs
+propose that the variational posterior can be expressed as an infinite mixture of distributions
+q(w|θ) =
+�
+q(w|z, θ)q(z|θ)dz
+(6)
+where θ is the learnable posterior parameter, and z ∼ q(z|θ) ≡ q(z)2 is a vector with the same
+dimension on the input layer, which plays the role of an auxiliary latent variable. Moreover, allowing local
+reparametrizations, the variational posterior for fully connected layers become a modification of Eq. 3
+written as
+w ∼ q(w|z) =
+�
+ij
+N(w; ziµij, σ2
+ij).
+(7)
+Notice that by enhancing the complexity of q(z), we can increase the flexibility of the variational posterior.
+This can be done using Normalizing Flows since the dimensionality of z is much lower compared to the
+weights. Starting from samples z0 ∼ q(z0) from fully factorized Gaussian Eq. 3, a rich distribution q(zK)
+can be obtained by applying a successively invertible K-transformations fK on z0
+zK = NF(z0) = fK ◦ · · · ◦ f1(z0);
+log q(zK) = log q(z0) −
+K
+�
+k=1
+log
+����det ∂fk
+∂zk−1
+���� .
+(8)
+Unfortunately, the KL divergence in Eq. 2 becomes generally intractable as the posterior q(w) is an
+infinite mixture as shown in Eq. 6. This is addressed also in (33) by evoking Bayes law q(zK)q(w|zK) =
+q(w)q(zK|w) and introducing an auxiliary distribution r(zK|w, φ) parameterized by φ, with the purpose
+of approximating the posterior distribution of the original variational parameters q(zK|w) to further lower
+bound the KL divergence term. Therefore, KL divergence term can be bounded as follows
+− KL
+�
+q(w)
+��p(w)
+�
+= −Eq(w)
+�
+log
+�q(w)
+p(w)
+��
+≥ −Eq(w)
+�
+log
+�q(w)
+p(w)
+�
++ KL
+�
+q(zK|w)
+��r(zK|w, φ)
+��
+= −Eq(w)
+�
+log
+�q(w)
+p(w)
+�
++ Eq(zK|w)
+�
+log
+� q(zK|w)
+r(zK|w, φ)
+���
+= −Eq(w)
+�
+Eq(zK|w)
+�
+log
+�q(w)
+p(w)
+��
++ Eq(zK|w)
+�
+log
+� q(zK|w)
+r(zK|w, φ)
+���
+= −Eq(w,zK)
+�
+log
+�q(w)
+p(w)
+�
++ log
+� q(zK|w)
+r(zK|w, φ)
+��
+= Eq(w,zK) [− log (q(w)q(zK|w)) + log r(zK|w, φ) + log p(w)] ⇒
+− KL
+�
+q(w)
+��p(w)
+�
+≥ Eq(w,zK)
+�
+− KL
+�
+q(w|zK)
+��p(w)
+�
++ log q(zK) + log r(zK|w, φ)
+�
+,
+(9)
+where we have taken into account that KL[P∥Q] ≥ 0, and the equality is satisfied iff P = Q. In the last
+line, the first term can be analytically computed since it will be the KL divergence between two Gaussian
+distributions, while the second term is given by the Normalizing flow generated by fK as we observe in
+2 The parameter θ will be omitted in this section for clarity of notation.
+5
+
+Hort´ua et al.
+Parameter estimation via BNNs
+Eq. 8. Finally, the auxiliary posterior term is parameterized by inverse normalizing flows as follows (34)
+z0 = NF−1(zK) = g−1
+1
+◦ · · · ◦ g−1
+K (zK);
+log r(zK|w, φ) = log r(z0|w, φ) +
+K
+�
+k=1
+log
+�����det ∂g−1
+k
+∂zk
+����� , (10)
+where one can parameterize g−1
+K as another normalizing flow. In the paper (32), the authors also propose a
+flexible parametrization of the auxiliary posterior as
+z0 ∼ r(zK|w, φ) =
+�
+i
+N(z0; ˜µi(w, φ), ˜σ2
+i (w, φ)).
+(11)
+We will use the parameterization of the mean ˜µ, and the variance ˜σ2 as in the original paper as well as the
+masked RealNVP (35) as choice of Normalizing flows.
+3
+N-BODY SIMULATIONS DATASET
+In this work, we leverage 2000 hypercubes simulation taken from The Quijote project (24). They
+have been run using the TreePM code Gadget-III (36), and their initial conditions were generated at
+z = 127 using 2LPT (37). The set chosen for this work is made of standard simulations with different
+random seeds with the intention of emulating the cosmic variance. Each instance corresponds to a three-
+dimensional distribution of the density field with size 643. The cosmological parameters vary according
+to Ωm ∈ [0.1, 0.5], Ωb ∈ [0.03, 0.07], h ∈ [0.5, 0.9], ns ∈ [0.8, 1.2], σ8 ∈ [0.6, 1.0], while neutrino mass
+(Mν = 0eV) and the equation of state parameter (w = −1) are kept fixed. The dataset was split into
+training(70%), validation (10%), and test (20%), while hypercubes were logarithmic transformed and the
+cosmological parameters normalized between 0 and 1. In this paper we will build BNNs with the ability to
+predict three out of five aforementioned parameters, Ωm, σ8 and h.
+4
+BNNS IMPLEMENTATION
+We will consider three different BNNs architectures based on the discussion presented in Section 2: standard
+BNNs (prior and variational posterior defined as a mean-field Normal distributions) [sBNNs]; BNNs with
+Flipout estimator [FlipoutBNNs]; and Multiplicative normalizing flows [VBNNs]. The experiments were
+implemented using the TensorFlow v:2.9 and TensorFlow-probability v:0.19 (38). All BNNs designed in this
+paper are comprised of three parts. First, all experiments start with a 643-voxel input layer corresponding to
+the normalised 3D density field followed by the fully-convolutional ResNet-18 backbone as it is presented
+schematically in table 1. All the Resblock are fully pre-activated and their representation can be seen
+in figure. 1. The repository Classification models 3D was used to build the backbone of BNNs (39).
+Subsequently, the second part of BNNs represents the stochasticity of the network. This is comprised
+of just one layer and it depends on the type of BNN used. For sBNNs, we employ the dense variational
+layer which uses variational inference to fit an approximate posterior to the distribution over both the
+kernel matrix and the bias terms. Here, we use as posterior and prior(no-trainable) Normal distributions.
+Experiments with FlipoutBNNs for instance, are made via Flipout dense layer where the mean field normal
+distribution are also utilized to parameterize the distributions. These two layers are already implemented in
+the package TF-probability (38). On the other hand, for VBNNs we have adapted the class DenseMNF
+implemented in the repositories TF-MNF, MNF-VBNN (32) to our model. Here, we use 50 layers for the
+masked RealNVP NF, and the maximum variance for layer weights is around the unity. Finally, the last
+6
+
+Hort´ua et al.
+Parameter estimation via BNNs
+ResNet-18 backbone
+Layer Name
+Input Shape
+Output Shape
+Batch Norm
+(Nbatch, 64,64,64,3)
+(Nbatch, 64,64,64,3)
+3D Convolutional
+(Nbatch, 70,70,70,3)
+(Nbatch, 32,32,32,64)
+Batch Norm+ReLU
+(Nbatch, 32,32,32,64)
+(Nbatch, 32,32,32,64)
+Max Pooling 3D
+(Nbatch, 34,34,34,64)
+(Nbatch, 16,16,16,64)
+Batch Norm+ReLU
+(Nbatch, 16,16,16,64)
+(Nbatch, 16,16,16,64)
+Resblock 1
+�
+(Nbatch, 16, 16, 16, 64)
+(Nbatch, 16, 16, 16, 64)
+�
+(Nbatch, 16,16,16,64)
+Batch Norm+ReLU
+(Nbatch, 16,16,16,64)
+(Nbatch, 16,16,16,64)
+Resblock 2
+�
+(Nbatch, 16, 16, 16, 64)
+(Nbatch, 8, 8, 8, 128)
+�
+(Nbatch, 8,8,8,128)
+Batch Norm+ReLU
+(Nbatch, 8,8,8,128 )
+(Nbatch, 8,8,8,128)
+Resblock 3
+�
+(Nbatch, 8, 8, 8, 128)
+(Nbatch, 4, 4, 4, 256)
+�
+(Nbatch, 4,4,4,256)
+Batch Norm+ReLU
+(Nbatch, 4,4,4,256 )
+(Nbatch, 4,4,4,256)
+Resblock 4
+�
+(Nbatch, 4, 4, 4, 256)
+(Nbatch, 2, 2, 2, 512)
+�
+(Nbatch, 2,2,2,512)
+Batch Norm+ReLU
+(Nbatch, 2,2,2,512 )
+(Nbatch, 2,2,2,512)
+Global Avg Pooling
+(Nbatch, 2,2,2,512)
+(Nbatch, 512)
+Table 1. Configuration of the backbone BNNs used for all experiments presented in this paper.
+part of all BNNs account for the output of the network, which is dependent on the aleatoric uncertainty
+parameterization. We use a 3D multivariate Gaussian distribution with nine parameters to be learnt (three
+means µ for the cosmological parameters, and six elements for the covariance matrix Σ).
+The loss function to be optimized during training is given by the ELBO 2 where the second term is
+associated to the negative log-likelihood (NLL)
+− NLL ∼ 1
+2 log |s · Σ| + 1
+2(y − µ)⊤ (s · Σ)−1 (y − µ),
+(12)
+averaged over the mini-batch. The scalar variable s is equal to one during the training process, and it
+becomes a trainable variable during post-training to recalibrate the probability density function (16, 31).
+The algorithm used to minimize the objective function is the Adam optimizer with first and second moment
+exponential decay rates of 0.9 and 0.999, respectively (40). The learning rate starts from 10−3 and it
+will be reduced by a factor of 0.8 in case that any improvement has not been observed after 10 epochs.
+Furthermore, we have applied warm-up period for which the model turns on progressively the KL term
+in Eq. 2. This is achieved by introducing a β variable in the ELBO, i.e., β · KL
+�
+q(w|θ)
+��p(w)
+�
+, so, this
+parameter starts being equal to 0 and grows linearly to 1 during 10 epochs (41). BNNs were trained with
+32 batches and early stopping callback for avoiding over-fitting. The infrastructure used was the Google
+Cloud Platform (GCP) using a nvidia-tesla-t4 of 16 GB GDDR6 in a N1 machine series shared-core.
+7
+
+Hort´ua et al.
+Parameter estimation via BNNs
+Figure 1a. Illustration of the first skip connection in
+a residual block.
+Figure 1b. Illustration of the second skip connection
+in the residual block.
+Figure 1. Each Resblock includes both skip connection configurations. (A) The Resblock starts with this
+configuration applied to the input tensor. (B) The output of the previous configuration is fed into this
+connection.
+4.1
+Metrics
+We compare all BNN results in terms of performance, i.e., the precision of their predictions for the
+cosmological parameters quantified through Mean Square Error (MSE), ELBO, and plotting the true vs
+predicted values with its coefficient of determination. Also, it is important to quantify the quality of the
+uncertainty estimates. One of the ways to diagnostic the quality of the uncertainty estimates is through
+reliability diagrams. Following (31, 11), we can define perfect calibration of regression uncertainty as
+Eˆσ2
+����
+E[(y − µ)2]
+�� ˆσ2 = α2�
+− α2���
+∀
+�
+α2 ∈ R
+�� α2 ≥ 0
+�
+.
+(13)
+Hence, the predicted uncertainty ˆσ2 is partitioned into K bins with equal width, and the variance per bin is
+defined as
+var(Bk) :=
+1
+��Bk
+��
+�
+i∈Bm
+1
+N
+N
+�
+n=1
+(µi,n − yi)2,
+(14)
+with N stochastic forward passes. On the other hand, the uncertainty per bin is defined as
+uncert(Bk) :=
+1
+|Bk|
+�
+i∈Bk
+ˆσ2
+i .
+(15)
+With these two quantities, we can generate reliability diagrams to assess the quality of the estimated
+uncertainty via plotting var(Bk) vs. uncert(Bk). In addition, we can compute the expected uncertainty
+8
+
+skip connection
+(identity)
+F() +
+Conv3D
+BN+ReLU
+Conv3D
++ (。)
+add
+output
+F()skip connection
+(identity)
+F() + 
+BN+ReLU
+Conv3D
+BN+ReLU
+Conv3D
++ (,)
+ppe
+output
+F()Hort´ua et al.
+Parameter estimation via BNNs
+Metrics
+FlipoutBNNs
+VBNNs
+sBNNs
+Ωm
+σ8
+h
+Ωmh2
+σ8Ω0.25
+m
+Ωm
+σ8
+h
+Ωmh2
+σ8Ω0.25
+m
+Ωm
+σ8
+h
+Ωmh2
+σ8Ω0.25
+m
+MSE
+0.063
+0.057
+0.190
+ELBO
+20.85
+19.71
+31.57
+r2
+0.82
+0.98
+0.2
+0.03
+0.93
+0.85
+0.99
+0.4
+0.56
+0.95
+0.75
+0.85
+0.01
+0.23
+0.80
+UCE
+0.109
+8.10
+0.26
+0.0008
+0.0008
+0.010
+>1.0
+Table 2. Metrics test set results for all BNNs architectures. High UCE values indicate miscalibration. MSE
+and ELBO are computed only over the cosmological parameters.
+calibration error (UCE) in order to quantify the miscalibration
+UCE :=
+K
+�
+k=1
+|Bk|
+m
+��var(Bk) − uncert(Bk)
+��,
+(16)
+with number of inputs m and set of indices Bk of inputs, for which the uncertainty falls into the bin k. A
+more general approach proposed in (16) consists in computing the expected coverage probabilities defined
+as the x% of samples for which the true value of the parameters falls in the x%-confidence region defined
+by the joint posterior. Clearly, this option is more precise since it captures higher-order statistics through
+the full posterior distribution. However, for simplicity, we will follow the UCE approach.
+5
+ANALYSIS AND RESULTS OF PARAMETER INFERENCE WITH BNNS
+In this section we discuss the results obtained by comparing three different versions of BNNs, the one
+with MNFs, the standard BNN, and the third one using Flipout as estimator. The results reported in this
+section were computed on the Test dataset. Table 2. shows the metrics obtained for each BNN approach.
+As mentioned, MSE, ELBO and r2 provide well estimates for determining the precision of the model,
+while UCE measures the miscalibration. Here, we can observe that VBNNs outperform all experiments,
+not only taking into account the average error, but also the precision for each cosmological parameter along
+with a good calibration in its uncertainty predictions. Followed by VBBNs, we have the FlipoutBNNs,
+however, although this approach yields good cosmological parameter estimation, it understimates their
+uncertainties. Therefore, VBNNs avoids indeed the application of an extra post training step in the Machine
+Learning pipeline related to calibration. Notice that in all experiments, h becomes hardly predicted for all
+model. Figure 2 displays the predicted against true values for Ωm, ωm (instead of h), σ8 and the degeneracy
+direction defined as σ8Ω0.25
+m . Error bars report the epistemic plus aleatoric uncertainties predicted by BNNs,
+which illustrates the advantages of these probabilistic models where the certainty prediction of the model is
+captured instead of traditional DNNs where only point estimates are present. This uncertainty was taken
+from the diagonal part of the covariance matrix.
+5.1
+Calibration metrics
+In figure 3, we analyze the quality of our uncertainty measurement using calibration diagrams. We show
+the predicted uncertainty vs observed uncertainty from our model on the Test dataset. Better performing
+uncertainty estimates should correlate more accurately with the dashed lines. We can see that estimating
+uncertainty from VBNNs reflect better the real uncertainty. Furthermore, the scale for VBNNs is two
+orders of magnitude lower than FlipoutBNN, which also implies how reliable is this models according
+to their predictions. Notice that the even if we partitioned the variance into K = 10 bins with equal
+width, FlipoutBNNs and sBNNs yield underestimate uncertainties (many examples concentrates in lower
+bin values), for this reason we see that while VBNNs supply all ten samples in the calibration plots, for
+9
+
+Hort´ua et al.
+Parameter estimation via BNNs
+Figure 2. Plots of True vs Predicted values provided by the best experiment VBNNs, for Ωm, σ8, and
+some derivative parameters. Points are the mean of the predicted distributions, and error bars stand for the
+heteroscedastic uncertainty associated to epistemic plus aleatoric uncertainty at 1σ.
+the others we have just 3-4 of them. Next, we employed the σ-scaling methodology for calibrating the
+FlipoutBNNs predictions (31). For doing so, we optimize uniquely the loss function described in Eq. 12
+where all parameters related to the BNNs where frozen, i.e., the only trainable parameter was s. After
+training, we got s ∼ 0.723, reducing UCE only up to 10%, and the number of samples in the calibration
+diagrams enlarged to 4-5. This minor performance enhancement means that σ-scaling is not suitable to
+calibrate all BNNs, and alternative re-calibration techniques must be taken into account in order to build
+reliable intervals. At this point, we have noticed the advantages of working with methods that leading with
+networks already well-calibrated after the training step (17).
+5.2
+Joint analysis for Cosmological parameters
+In order to show the parameter intervals and contours from the N-body simulations, we choose randomly
+an example from the test set with true values shown in table 3. The two-dimensional posterior distribution
+of the cosmological parameters are shown in figure 4 and the parameter 95% intervals are reported in
+table 3. We can observe that VBNNs provides considerably tighter and well constraints on all parameters
+10
+
+0.6
+perfectmatch
+1.1
+perfectmatch
+0.5
+1.0
+8
+0.9
+Predicted
+Predicted
+0.3
+0.8
+0.2
+0.7
+0.1
+0.6
+0.2
+0.3
+0.4
+0.7
+0.8
+0.9
+True08
+perfect match
+perfect match
+0.40
+0.8
+0.35
+0.7
+ywu
+0.30
+0.25
+Predicted
+0.6
+0.20
+0.5
+0.15
+0.10
+0.4
+0.05
+0.4
+0.5
+0.6
+0.7
+0.8
+0.1
+0.2
+0.3
+True Ωmh?Hort´ua et al.
+Parameter estimation via BNNs
+Figure 3. Calibration diagrams for the best experiments, VBNNs and FlipoutBNNs. The lower is the
+UCE value, the higher is the calibration of the model. Dashes lines stand for the perfect calibration, so, the
+discrepancy to this identity curve reveals miscalibration.
+with respect to the sBNNs (18). Most important, this technique offers also the correlation among parameters
+and the measurement about how reliable the model in their predictions.
+6
+CONCLUSIONS
+N-body simulations offer one of the most powerful ways to understand the initial conditions of the
+Universe and improve our knowledge on fundamental physics. In this paper we used QUIJOTE dataset, in
+order to show how convolutional DNNs capture non-Gaussian patters without requiring a specifying the
+summary statistic (such as PS). Additionally, we have show how we can build probabilistic DNNs to obtain
+uncertainties which account for the reliability in their predictions. One of the main goals of this paper was
+11
+
+CalibrationforQm withVBNN
+Calibrationforo:withVBNN
+1e-3
+1e-3
+UCE=0.0008
+UCE=0.0008
+4
+2
+m
+2
+1
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Expecteduncertainty
+1e-3
+Expecteduncertainty
+1e-3
+le-2 Calibration for h with VBNN
+Calibration for Qm with FlipoutBNNs
+0.3
+rtainty
+2.5
+UCE=0.0105
+Observed uncertainty
+UCE=0.1095
+2.0
+uncer
+0.2
+1.5
+Observed
+0.1
+1.0
+0.5
+0.0
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Expected uncertainty
+1e-2
+Expecteduncertainty
+CalibrationforO: withFlipoutBNNs
+CalibrationforhwithFlipoutBNNs
+Observed uncertainty
+20
+UCE=8.099
+UCE=0.2595
+15
+4
+3
+10
+2
+5
+1
+0
+5
+10
+15
+20
+0
+1
+2
+3
+4
+5
+Expected uncertainty
+ExpecteduncertaintyHort´ua et al.
+Parameter estimation via BNNs
+Figure 4. 68% and 95% parameter constraint contours from one example of Quijote test dataset using
+VBNNs and FlipoutBNNs. The diagonal plots are the marginalized parameter constraints, the dashed lines
+stand for the the true values. This plot was made using Getdist (42).
+also reporting how improves these BNNs when we integrate them with techniques such as a Multiplicative
+normalizing flows to enhance the variational posterior complexity. We found that VBNNs not only provides
+considerably tighter and well constraints on all cosmological parameters as we observed in figure 4, but
+also yields with well-calibrated estimate uncertainties as it was shown in figure 3. Nevertheless, some
+limitations in this research includes simple prior assumptions (mean-field approximations), lower resolution
+in the simulations, and absence of additional calibration techniques. These restrictions will be analysed in
+detail in a future paper.
+12
+
+0.6
+D
+0.5
+m
+0.4
+0.3
+0.8
+0o 0.7
+b
+0.6
+1.0
+60.8
+0.6
+0.65
+.25
+0.60
+0.55
+0.50
+0.6
+2
+0.4
+0.2
+0.3
+0.4
+0.5
+0.6
+0.6
+0.7
+0.8
+0.6
+0.8
+1.0
+0.50 0.55 0.60 0.65
+0.2
+0.4
+0.6
+Qm
+h
+0:Q0.25
+Qmh2
+m
+VBNNs
+FlipoutBNNsHort´ua et al.
+Parameter estimation via BNNs
+Parameter
+95% limits VBNNs
+95% limits FlipoutBNNs
+True Value
+Ωm
+0.47+0.10
+−0.10
+0.45+0.11
+−0.11
+0.495
+σ8
+0.697+0.038
+−0.038
+0.699+0.059
+−0.060
+0.699
+h
+0.81+0.17
+−0.17
+0.78+0.20
+−0.19
+0.800
+σ8Ω0.25
+m
+0.577+0.051
+−0.052
+0.573+0.063
+−0.064
+0.587
+Ωmh2
+0.31+0.19
+−0.18
+0.573+0.063
+−0.064
+0.317
+Table 3. Parameter 95% intervals taken from the parameter constraint contours (figure 4) from one example
+of Quijote test dataset using VBNN and FlipoutBNN.
+ACKNOWLEDGMENTS
+This paper is based upon work supported by the Google Cloud Research Credits program with the award
+GCP19980904.
+Leonardo Casta˜neda was supported by patrimonio aut´onomo fondo Nacional de financiamiento para
+la ciencia y la tecnolog´ıa y la innovacion Francisco Jos´e de Caldas (Minciencias Colombia) grant No
+110685269447 RC-80740-465-2020 projects 69723. H. J. Hort´ua acknowledges the support from cr´editos
+educaci´on de doctorados nacionales y en el exterior- colciencias, and the grant provided by the Google
+Cloud Research Credits program.
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+

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+Lieb lattices and pseudospin-1 dynamics under barrier- and
+well-like electrostatic interactions
+V. Jakubsk´y1 and K. Zelaya1
+1Nuclear Physics Institute, Czech Academy of Science, 250 68 ˇReˇz, Czech Republic
+Abstract
+This work considers the confining and scattering phenomena of electrons in a Lieb lattice
+subjected to the influence of a rectangular electrostatic barrier.
+In this setup, hopping
+amplitudes between nearest neighbors in orthogonal directions are considered different, and
+the next-nearest neighbor interaction describes spin-orbit coupling. This makes it possible to
+confine electrons and generate bound states, the exact number of which is exactly determined
+for null parallel momentum to the barrier. In such a case, it is proved that one even and one
+odd bound state is always generated, and the number of bound states increases for non-null
+and increasing values of the parallel momentum. That is, bound states carry current. In the
+scattering regime, the exact values of energy are determined where the resonant tunneling
+occurs. The existence of perfect tunneling energy in the form of super-Klein tunneling is
+proved to exist regardless of the bang gap opening. Finally, it is shown that perfect reflection
+appears when solutions are coupled to the intermediate flat-band solution.
+1
+Introduction
+The theoretical and experimental progress in the physics of graphene and other Dirac materials
+has become a trending topic in material science and theoretical physics [1,2]. Many remarkable
+properties of these materials follow from the fact that dynamics of low-energy quasi-particles is
+described by equations known in relativistic quantum mechanics. It makes it possible to test
+relativistic properties such as Klein tunneling [3,4], relativistic Landau levels, and the existence
+of pseudoparticles violating the Lorentz invariance [5, 6] (type-II Dirac fermions). Graphene
+mono- and multi-layer systems exhibit transport properties such as quantum Hall effect [7]
+and anomalous quantum Hall effect in graphene [8], and Josephson effect in twisted cuprate
+bilayers [9].
+Graphene has shown to be a helpful benchmark system to test the properties of relativistic
+pseudospin-1/2 particles in low-energy systems.
+Nevertheless, the family of Dirac materials
+contains also other, equally interesting, members.
+Their geometries can extend beyond the
+honeycomb lattice.
+For instance, there are Kagome [10], Dice or α − T3 [11, 12], and Lieb
+lattices [13, 14], which lead to effective pseudospin-1 Dirac equations. It was recently showed
+that the Kagome lattice can be obtained from a geometrical deformation of the Lieb lattice [15].
+For a recent survey of two-dimensional lattices and their physical properties and realization,
+see [16].
+1
+arXiv:2301.03552v1  [cond-mat.mes-hall]  9 Jan 2023
+
+Particularly, the Lieb lattice is a two-dimensional array with a periodicity of a square lattice.
+The sites are located in the corners of each square and at the midpoints on its sides. To our
+best knowledge, the Lieb lattice has not been found in nature. However, it has been prepared
+artificially in diverse ways [17, 18]. It was realized in experiments with optical fibers [19–23].
+Furthermore, it was formed by ultracold atoms trapped in optical lattices [24] or by electrons
+of Cu(111) atoms confined by an array of CO molecules [25]. It was also prepared in covalent-
+organic frameworks [26].
+The tight-binding model can well describe the band structure of the Lieb lattice. It reveals
+the existence of two bands with positive and negative energies and an additional so-called flat
+band. The latter is associated with the states that have fixed (zero) energy independent of the
+value of momentum. It is worth mentioning that the flat band solutions were prepared in the
+optical experiments, see [22, 23]. Similarly to graphene, the dynamics of the low-energy quasi-
+particles in the Lieb lattice is dictated by a relativistic Dirac-type equation. Nevertheless, these
+quasi-particles have pseudospin-1 due to three atoms per unit cell.
+In the current article, we investigate the scattering and confinement of the relativistic quasi-
+particles by a rectangular electric potential in the Lieb lattice with a gapped band structure.
+Gap-opening can be induced by on-site energy that differs on three sublattices or by the phase
+acquired by the electron when jumping between the neighboring sites [24], see also [27]. In
+the article, we adopt the second approach where a purely imaginary next nearest-neighbor
+interaction, attributed to spin-orbital coupling [13], is taken into account.
+Effects such as electron confinement and transmission are obtained with the aid of the proper
+boundary conditions, which enforce the continuity on two out of the three pseudospin-1 compo-
+nents. The third component can be discontinuous, which leads to a spatial discontinuity in the
+probability density. Nevertheless, it does not compromise the associated continuity equation.
+Electron dynamics for electrostatic interactions in graphene have been discussed in the litera-
+ture, such as the transmission properties in square barriers [28,29] and electron confinement with
+cylindrical quantum dots [30]. We thus focus on the related properties of the quasi-particle dy-
+namics in the Lieb lattice. We further analyze the influence of the flat-band solution in electron
+dynamics. As shown in the manuscript, solutions in this regime are described by degenerate
+Bloch-wave solutions whose linear combinations can compose wavepackets of arbitrary form.
+These are shown to be current-free solutions regardless of the nature of the wavepacket. As a
+result, one obtains perfectly reflected waves when they couple to flat-band solutions.
+The manuscript is structured as follows. In Sec. 2 we briefly introduce and discuss the main
+properties of the Lieb lattice with nearest next-nearest neighbor interactions, from which the
+effective low-energy Dirac equation is obtained.
+In Sec. 3, we present the general solutions
+and the transfer matrix associated with the rectangular electrostatic interaction. The latter
+is then exploited in Sec. 4 and Sec. 5 to discuss in full detail the localization of electrons and
+scattering dynamics, respectively. Finally, discussions and perspectives are provided in Sec. 7,
+and complementary details about the proof of the number of bound states are given in App. A.
+2
+
+(a)
+(b)
+Figure 1: (a) Lieb lattice, composed by the atoms A (blue-filled circle), B (green square), and C
+(red-filled square). The dashed arrows denote the direction of positive phase hopping parameter
+between next-nearest neighbors B − C. (b) Composition of a unit cell of the Lieb lattice. The
+unit displacement vectors ⃗δ1 = aˆx and ˆδ2 = aˆy connect the atom A with B and A with C,
+respectively. The corresponding nearest hopping parameters are t1, t2, whereas the next-nearest
+neighbor hopping parameter is +it3 and −it3 depending if it occurs in the direction denotes by
+the arrows.
+2
+Lieb lattice and pseudospin-1 Dirac equation
+Let us consider an electronic Lieb lattice 1 so that the separation between two nearest atoms is
+a, the length of each side of the square is ℓ = 2a. There are three sites in the elementary cell, see
+Fig. 1a. The primitive translation vectors are ⃗r1 = 2aˆx and ⃗r2 = 2aˆy. It is customary to denote
+the atoms at the corners of the square as A, whereas the atoms at the sides of the square are B
+(horizontal) and C (vertical). The lattice vectors ⃗δ1 = aˆx = ⃗r1/2 and ⃗δ2 = aˆy = ⃗r2/2 connect an
+atom on the site A to those on the sites B and C, respectively (see Fig. 1b). The atoms A, B and
+C form the three sublattices RA = n1⃗r1 +n2⃗r2, RB = ˜RA +⃗δ1, and RC = ⃗RA +⃗δ2, respectively,
+with n1, n2 ∈ Z. The reciprocal space is spanned by the translation vectors of the reciprocal
+space ˆrk1 and ˆrk2, ˆrp · ˆrkq = 2πδp,q, p, q = 1, 2. This leads to ˆrk1 = π
+a ˆx and ˆrk2 = π
+a ˆy. The
+first Brillouin zone, constructed from the Wigner-Seitz rule, restricts to the region composed by
+kx ∈ [− π
+2a, π
+2a] and ky ∈ [− π
+2a, π
+2a].
+The band structure of the electrons on the Lieb lattice can be analyzed with the use of the
+tight-binding model. There are considered the nearest neighbor (NN) interactions between the
+sites A − B and A − C, represented by the hopping parameters t1 and t2, respectively. We take
+into account also the next-nearest neighbor (NNN) transition B − C, which can be complex
+valued, with the sign of phase dependent on the orientation of the hopping. This emerges due
+to external time-dependent driven fields in photonic Lieb lattices [31], and magnon Lieb and
+Kagome lattices [32].
+In particular, we consider a purely imaginary NNN hopping parameter e±iπ/2t3, where the
+1The results here obtained apply to optical Lieb lattices as well.
+3
+
+C
+A
+BA
+B(a) t3 = 0
+(b) t3 ̸= 0
+Figure 2: Dispersion bands w+(⃗k) (yellow-upper), w−(⃗k) (green-lower), and w0(⃗k) (blue-middle)
+for the gapless (a) and gapped (b) configurations.
+hopping phase is positive (+) is the hopping occurs counter-clock-wise, and negative (−) oth-
+erwise. Such a hopping dynamics is depicted in Fig. 1a. This type of hopping was introduced
+by Haldane in [8] as a model for quantum anomalous Hall effect in graphene without strong
+external magnetic fields, which was latter found experimentally in [33]. See also [34] for a recent
+review.
+The spectral analysis of the tight-binding Hamiltonian reveals that there are three bands in
+its spectrum [13],
+w0(⃗k) = 0,
+w±(⃗k) = ±2
+�
+t2
+1 cos2(akx) + t2
+2 cos2(aky) + 4t2
+3 sin2(akx) sin2(aky).
+(1)
+The bands have linear dependence on the momentum in the four Dirac points that are
+situated in the first Brillouin zone. Their explicit position depends on the relative strength of
+t3. In this work, we focus on the most relevant situation where t3 < t1
+2 , t3 < t2
+2 . In that case,
+the Dirac point is ⃗K = ( π
+2a, π
+2a), see Fig.2b for illustration. A similar analysis holds for higher
+values of t3, where the Dirac points are displaced with respect to ⃗K. For a detailed discussion,
+see [13].
+Let us calculate the approximate form of the tight-binding Hamiltonian in the vicinity of the
+Dirac point ⃗K. We denote the effective operator as H(⃗k) ≡ H( ⃗K + ⃗k), where |⃗k| is considered
+small enough so that we can keep terms up to first-order in ⃗k. The proper expansion of H(⃗k)
+at the Dirac point ⃗K can be conveniently written as
+H(⃗k) = 2at1kxS1 + 2at2kyS2 + 4t3S3.
+(2)
+The matrices
+S1 =
+�
+�
+0
+1
+0
+1
+0
+0
+0
+0
+0
+�
+� ,
+S2 =
+�
+�
+0
+0
+1
+0
+0
+0
+1
+0
+0
+�
+� ,
+S3 =
+�
+�
+0
+0
+0
+0
+0
+−i
+0
+i
+0
+�
+� ,
+(3)
+form the three-dimensional representation of su(2) algebra, [Sp, Sq] = iεpqrSr, with εpqr the
+three-dimensional anti-symmetric tensor. Therefore, the quasi-particles described by the effec-
+tive Hamiltonian (2) have pseudospin 1.
+4
+
+元
+0
+2a
+a
+2
+w(k)
+0
+-2
+0
+元
+2a
+ky
+a元
+0
+2a
+a
+2
+w(k)
+-2
+0
+元
+2a
+ky
+aIt is worth noting that, for t3 = 0, the resulting Dirac Hamiltonians in (2) becomes linear
+combinations of the spin-1 matrices S1 and S2. In such a case, the matrix �S,
+�S =
+�
+�
+−1
+0
+0
+0
+1
+0
+0
+0
+1
+�
+� ,
+(4)
+satisfies {�S, Sj} = 0, with j = 1, 2, and represents the chiral symmetry of H as there holds
+{�S, H|t3=0} = 0. The later relation implies that the eigenvalues E of H|t3=0 are symmetric
+with respect to E = 0. When an eigenstate ΨE of H has energy E, then there is an eigenstate
+Ψ−E = �SΨE with the energy of the opposite sign.
+2.1
+External electrostatic interaction
+Throughout this manuscript, we consider a piece-wise continuous external electric field dis-
+tributed in the ˆx direction, while we discard any magnetic interaction. The corresponding effec-
+tive Hamiltonian is obtained from (2) through the Peierls transformation [35,36], ⃗k → −iℏ⃗∇ and
+iℏ∂t → iℏ∂t − U(⃗x)I, with I the 3 × 3 identity matrix. Since the Hamiltonian becomes invariant
+on the ˆy direction, the eigenstates can be cast in the form Ψ(x, y) → e±ik2yΨ(x), where Ψ(x)
+solve the following stationary equation:
+H(x)Ψ(x) = (−iℏv1S1∂x + ℏv2kyS2 + mS3 + Ua I)Ψ(x) = EΨ(x),
+(5)
+with Ψ(x) = (ψA(x), ψB(x), ψC(x))T .
+In (5), we have used v1 = 2at1, v2 = 2at2 and m = 4t3 to simplify the notation. This allows
+us relating v1 and v2 to the Fermi velocities along the ˆx and ˆy directions, respectively, whereas
+m plays the role of the mass term in the Dirac equation. Furthermore, we have considered
+a constant electrostatic potential, which is valid for our purposes since we are dealing with
+piece-wise continuous interactions.
+From the previous considerations, we may decouple the eigensolution components ψA,B,C as
+follows:
+− ℏ2v2
+1ψ′′
+A + ℏ2v2
+2k2
+yψA = ((E − Ua)2 − m2)ψA,
+(6)
+ψB = −iℏv1(E − Ua)ψ′
+A + ℏmv2kyψA
+(E − Ua)2 − m2
+,
+ψC = ℏmv1ψ′
+A + ℏv2ky(E − Ua)ψA
+(E − Ua)2 − m2
+,
+(7)
+where the hopping parameters tj, for j = 1, 2, 3.
+The probability current associated with Ψ can be calculated in standard manner from the
+continuity equation ∂tρ + ⃗∇ · j = 0. Here, ρ = Ψ†Ψ stands for the probability density, and the
+probability current takes the form
+j = (2v1 Re ψ∗
+AψB, 2v2 Re ψ∗
+AψC) .
+(8)
+Let us consider briefly the situation when the potential has a finite discontinuity at x = x0.
+It is necessary to specify the behavior of the wave functions at this point. It can be done by
+integrating (5) in the vicinity of x0. Alternatively, one can require the component of the density
+5
+
+current perpendicular to the barrier to be continuous. The second approach is more general and
+covers the boundary conditions provided by the integration as the special case that read as
+ψA(x−
+0 ) = ψA(x+
+0 ),
+ψB(x−
+0 ) = ψB(x+
+0 ).
+(9)
+It is worth noting that only two of the three eigensolution components are required to be
+continuous in x0, and the third component ψC can have a discontinuity at this point.
+The
+corresponding probability density is not necessarily continuous. This observation was made in
+pseudospin-1 photonic lattices [37]. The boundary conditions obtained in (9) keep the current
+of probability density in the ˆx direction continuous, which is the component perpendicular to
+the discontinuity. As the component ΨC(x) can be discontinuous at x0, the tangent current and
+the probability densities are not necessarily continuous.
+3
+Rectangular electrostatic barrier
+Let us consider an external electrostatic electric potential homogeneous along the ˆy direction
+and piece-wise continuous across the ˆx direction, with
+U(x) =
+�
+0
+|x| > L
+2
+U0
+|x| ≤ L
+2
+.
+(10)
+We consider, without loss of generality, U0 > 2m. Solutions of the stationary equation are split
+into three regions, namely, the region I (x < L/2), region II (−L/2 ≤ x ≤ L/2), and region III
+(x > L/2). The latter are written as
+��ka = eikyyeikax
+�
+�
+�
+1
+−iℏmv2ky+ℏv1ka(E−Ua)
+(E−Ua)2−m2
+iℏmv1ka+ℏv2ky(E−Ua)
+(E−Ua)2−m2
+�
+�
+� ,
+a = I, II, III,
+(11)
+where UI = UIII = 0, UII = U0, and consequently kI = kIII.
+We consider ky is fixed as a
+real quantity to obtain plane-wave solutions propagating parallel to the barrier. In turn, ka is
+considered a complex parameter so that we can distinguish two different regimes (see discussion
+below).
+The solution (11) satisfies the eigenvalue equation (5) with the eigenvalue
+E = Ua ±
+�
+m2 + ℏ2(v2
+1k2a + v2
+2k2y) = Ua ±
+�
+˜m2 + ℏ2v2
+1k2a,
+(12)
+where we have introduced the effective mass term
+�m =
+�
+m2 + ℏ2v2
+2k2y.
+(13)
+From (11), we distinguish two behaviors, namely, plane-wave solutions for ka ∈ R and
+evanescent-wave solutions for ka = −ipa, with pa ∈ R.
+In both cases, the wave functions
+are associated with real eigenvalues. They are classified as
+ka ∈ R,
+E(ka, ky) = Ua ±
+�
+�m + v2
+1ℏ2k2a,
+E(pa, ky) ∈ (−∞, U0 − �m) ∪ (Ua + �m, ∞),
+(14)
+ka = −i pa,
+E(pa, ky) = Ua ±
+�
+�m − v2
+1ℏ2p2a,
+E(pa, ky) ∈ (Ua − �m, U0 + �m).
+(15)
+6
+
+(a)
+(b) |E| > m
+(c) |E| < m
+Figure 3: (a) Sketch of the energy surfaces spanned by the dispersion relations (14) (orange)
+and (15) (blue), together with two energy planes located at arbitrary energies |E| > m and
+|E| < m. Panel (b) and panel (c) depict the contour plot generated by the interception between
+the dispersion relations and the energy planes |E| > m and |E| < m, respectively. In panel (b),
+ξ = arctan(ky/kI) denotes the incidence angle of the plane wave and ky;c =
+√
+E2 − m2/ℏv2 the
+critical value of ky separating the evanescent-wave and plane-wave regimes.
+These dispersion relations span paraboloid and hyperboloid surfaces for plane-wave and evanescent-
+wave solutions, respectively. This behavior is depicted in Fig. 3a for UI = 0 (case a = I).
+For |E| > m, the behavior of the solutions is classified according to the values of ky, with
+ky;c =
+√
+E2 − m2/ℏv2 being the critical value. That is, for |ky| < ky;c, the solutions are plane-
+wave-like and the momenta kI and ky span an elliptic curve for a fixed energy. For |ky| > ky;c,
+the solutions become evanescent waves and pI and ky span a hyperbolic curve for the same fixed
+energy. This is sketched in Fig. 3b. For |E| < m, no plane-wave solutions exist for ky ∈ R,
+and only evanescent-wave solutions are generated. Here, pI, ky span a rotated hyperbola with
+respect to the case |E| > m, as depicted in Fig. 3c.
+The partial solutions Ξka at the regions I, II, III have to be combined in order to comply
+with the boundary conditions (9) at x0 = ±L and |x| → ∞. The wave function takes the general
+form
+Ξa(x, y) = αaΞ±
+ka(x, y) + βaΞ±
+−ka(x, y),
+a = I, II, III .
+(16)
+The boundary conditions (9) impose the continuity of the two upper components of the wave
+function, from which we find the set of relations between the coefficients αI, βI and αIII, βIII.
+That is,
+M
+�αI
+βI
+�
+=
+�αIII
+βIII
+�
+,
+M =
+�m11
+m12
+m21
+m22
+�
+,
+(17)
+with M being the transfer matrix, whose elements mij are functions of E, ky, m and U0. The
+7
+
+El>m
+E(k,k2)
+Ek<m?
+Plane-wave
+<region
+K
+Evanescentwave
+K=ki
+region
+K=iP1(p1,k2)
+k2;c
+(k1,k2)
+-K2:c
+K(p1,k2)
+Klatter are explicitly given by
+m11 = eiLkI
+�
+cos(LkII) − i sin(LkII)2v2
+1ℏ2k2
+I k2
+II + ˜m2(k2
+I + k2
+II)
+2E(E − V )kIkII
+�
+,
+(18)
+m22 = e−iLkI
+�
+cos(LkII) + i sin(LkII)2v2
+1ℏ2k2
+I k2
+II + ˜m2(k2
+I + k2
+II)
+2E(E − V )kIkII
+�
+,
+(19)
+m12 = isin(LkII)(k2
+yv2
+2(m2 + ℏ2k2
+yv2
+2) − v2
+1(m2 − ℏ2k2
+yv2
+2)k2
+1 − 2iv1v2kIkyE)( U0
+2 − E)
+ℏ2v2
+1kIkII(E2 − m2)(E − U0)
+,
+(20)
+where kI =
+√
+E2 − ˜m2/ℏv1 and kII =
+�
+(E − V )2 − ˜m2/ℏv1.
+The determinant of the transfer matrix is equal to one, in coherence with conservation of
+the probability current at the boundary. When kI and kII are real, there also holds m11 = m∗
+22
+and m12 = m∗
+21. In the next section, we shall use the transfer matrix for determinantion of the
+bound state energies as well as of the scattering characteristics of the plane-wave solutions.
+Additionally to (11), the Lieb lattice supports an additional solution in the form of a flat
+band, which is depicted in Fig. 2. This appears whenever E = Ua, and the eigensolutions cannot
+be determined from Eq. (6)-(7). Instead, one shall solve the Dirac Hamiltonian for E = Ua,
+which leads to the eigensolution Ξfb = (mχ, iℏv2kyχ, −ℏv1χ′)T , where χ = χ(x) is an arbitrary
+complex-valued function. Such an indeterminacy is better understood if one chooses χ such that
+Ξfb(ν, x) = eikyyeiνx
+�
+�
+m
+iℏv2ky
+−iℏv1ν
+�
+� ,
+(21)
+which is a flat band eigensolution for any ν ∈ C. Particularly, for ν ∈ R, Eq. (21) form a set of
+degenerate plane-wave solutions, usually known as degenerate Bloch waves [38] (see also Sec. 2.1
+in [18]). These degenerate waves form a continuous basis that can be used to construct arbitrary
+wavepackets through Fourier transforms. The latter has been exploited to construct the so-called
+compact localizes states [39], which are specific linear combinations of degenerate waves localized
+in each unitary cell of a finite-dimensional lattice. See [18, 40] for a more extensive discussion
+on the matter.
+It is clear that degenerate Bloch waves do not carry current on the x-direction, as jx =
+2v1 Re ψ∗
+AψB vanishes for any ν ∈ C.
+This also holds for any linear combination (finite or
+infinite) of degenerate Bloch state. Thus, the current states belonging to the flat band energy
+are current-free states.
+Notice that the dispersion and flat bands have a touching point only for m = 0 (See Fig. 2a),
+and thus one can explore the behavior of the solutions on the dispersion band when they approach
+the flat band interception. It is straightforward to realize that Ξka,ky,m→0 leads to the null vector,
+which is only one of the infinitely many solutions inside the flat band. For this reason, we shall
+discuss the flat band and the dispersion bands separately.
+4
+Electron confinement
+Let us explore the possibility of bound states trapped by the electrostatic potential (10). Here,
+we look for eigenvalues E so that the corresponding eigensolutions have finite norm in L2 ⊗ C3,
+8
+
+(a) ky = 0
+(b)
+(c)
+Figure 4: Sketch for the energy configuration associated with (10) for ky = 0 (a) and increasing
+values of ky (b)-(c). The diagonal-pattern and color-shaded regions denote the area covered by
+mass term m and effective mass term �m, respectively. In the panel (b), the energy (red-dashed
+line) inside the region II lies out of the effective mass term (plane-wave solution), whereas in
+the panel (c) they lie inside the effective mass term (evanescent-wave solution).
+which implies that eigensolutions must decay asymptotically to zero in the regions I and III for
+x → −∞ and x → ∞, respectively.
+Following (11), we thus use evanescent-wave solutions for the regions I and III. By fixing
+kI = kIII = ipI,
+pI > 0,
+(22)
+one restricts the energies into the interval E ∈ (− �m, �m), as depicted in all the cases of Fig. 4.
+The wave function composed from (16) has an exponentially vanishing behavior for |x| → ∞.
+This implies that we fix αI = 0, βI = 1 and βIII = 0, and the relation (17) turns into
+m12 = αIII,
+m22 = 0.
+(23)
+The first relation determines the amplitude of the wave function in the region III, whereas
+the second relation fixes the energies for the bound states. This can be written, after some
+simplifications, in the following form:
+tanh
+��
+�m2 − (E − U0)2
+ℏv1
+L
+�
+= −E(E − U0)
+�
+�m2 − (E − U0)2√
+�m2 − E2
+(E − U0)2( �m2 − E2) + �m2U0
+�
+E − U0
+2
+� .
+(24)
+The wave function ΞII in the intermediate region II can be either oscillatory for (E − U0)2 >
+˜m2 (we can set kII > 0 without loss of generality) or evanescent for (E − U0)2 < ˜m2 (kII = i pII,
+pII > 0), see Fig. 4b and Fig. 4c, respectively.
+The transcendental equation (24) allows us
+determining the bound state energies as a function of ky for both cases.
+Although the explicit solution E = E(ky) of (24) has to be found numerically, some pre-
+liminary information can be extracted by considering large values ℏv2ky ≫ U0, m in the tran-
+scendental equation (24). Here, �m ≈ ℏv2ky and the dispersion relation reduces to E2 ≈ E2
+∞ =
+ℏ2(−v2
+1p2
+I + v2
+2k2
+y). Since pI should be a real quantity in order to remain in the evanescent-wave
+regime in the regions I and III, we find that ℏv2|ky| ≥ E(ky) holds for asymptotic values of
+ℏv2ky. The behavior of E(ky) is thus bounded for ℏv2ky → ∞ and can be classified into the
+following in three asymptotic cases:
+9
+
+Uo+m
+Uo+m
+Uo
+m
+Uo-m
+10%m/7/7
+0
+-m
+1
+II
+III
+-mJo+m
+Uo
+Uo-m
+m 
+E
+0
+-m
+II
+IIIUo+m
+Uo
+Uo-m
+Uo-m
+m
+m
+E
+0
+-m
+-m
+II
+III(a)
+(b)
+Figure 5: (In units of ℏ=1) (a) Bound state energies E(ky), computed from (24), as a function
+of the transverse momentum ky for v1 = v2 = L = 1, m = 0.5, and U0 = 1.5. The blue-solid and
+red-dashed curves indicate bound state energies for arbitrary ky, whereas green-dot-dashed and
+black-dotted curves are energies emerging from a specific ky ̸= 0. The shaded area marks the
+scattering-state energy region. (b) Current parallel to the barrier Jy = ∂E(ky)/∂ky associated
+with the dispersion relations in (a).
+• First, a valid asymptotic behavior may be of the form E(ky → ∞) → C < ∞. Substituting
+the latter into (24) leads to a unique solution of the form E(ky → ∞) → C = U0/2.
+• Another possible asymptotic behavior is |E(ky)| = ℏv2|ky|, which vanishes both sides
+of (24). That is, |E(ky)| = ℏv2|ky| is a valid asymptotic behavior.
+• The last possible asymptotic behavior is |E(ky)| < ℏv2|ky|, which leads to a contradiction
+once substituted into (24). That is, such an asymptotic behavior is do not generate bound
+state solutions.
+We thus conclude that the eigenvalues associated with bound states, if they exist, either
+converge asymptotically to U0/2 or ℏv2|ky|. Since the current density on the direction parallel
+to the barrier is Jy = ∂E(ky)/∂ky ≡
+�
+R j(x, y)dx (see [41] or Appendix E in [42]), it converges
+either to zero or ±ℏv2 for ky → ∞.
+As an illustrative example, let us consider numerical values such that we have homogeneous
+Fermi velocities v1 = v2 = 1, a mass term m = 0.5, together with a rectangular potential well
+with L/ℏ = 1 and U0 = 1.5. Numerical solutions of (24) reveal the existence of two bound
+states for ky = 02, and new bound states appear for increasing values of ky. This is depicted in
+Fig. 5a, where one may see that energies indeed converge to either U0
+2 = 0.75 or become linear
+in ℏv2ky for large enough ky. Likewise, we depict in Fig. 5b the corresponding current density
+parallel to the barrier (Jy), which becomes finite or null for asymptotic ky, as predicted from
+our former analysis.
+• Further information is available for direct incidence, that is, ky = 0, �m = m. Here, the
+effective Hamiltonian possesses the additional symmetry represented [H, Px �S] = 0, with Px is
+the parity operator and �S defined in (4). This allows establishing a parity-symmetric criteria for
+the wave function �Ξ with respect to Px �S, namely, we classify the solutions fulfilling the condition
+Px �SΞ = ±Ξ as even (Ξ(e) for +) and odd (Ξ(e) for −). In this form, the coefficients of ΞII
+2This result agrees with the analytic formula presented in (26).
+10
+
+2
+0
+-4
+-2
+2
+4
+k2(k2
+-8
+-4
+4
+8
+K(a) U0 = 1.5
+(b)
+(c)
+Figure 6: (In units of ℏ=1) (a) Number of even (dotted) and odd (blue-thick) bound states
+as a function of L for v1 = 1, m = 0.5 and U0 = 1.5. (b) Eigensolution component ψC and
+(E − U(x))ψC for L = v1 = 1 and U0 = 1.5 and the even bound state energy E ≈ 0.281398.
+(c) Probability distribution associated with the eigenvalues E ≈ 0.281398 (blue-solid) and E ≈
+−0.32653 (red-dashing) and the same parameters as in (b).
+in (16) are αII = ±βII for even (+) and odd (−) functions, so that after evaluating the boundary
+condition at x = L one obtains relations to determine the energies of even and odd states as
+tan
+�kIIL
+2
+�
+= F(E),
+− cot
+�kIIL
+2
+�
+= F(E),
+F(E) =
+E
+U0 − E
+�
+(E − U0)2 − m2
+m2 − E2
+,
+(25)
+respectively, with kII =
+�
+(E − U0)2 − m2/ℏv1.
+Although the exact values of E cannot be analytically determined for arbitrary L, one can
+still determine the exact number of even (N(e)) and odd (N(o)) bound states. The thorough
+analysis (see App. A for a detailed proof) leads to
+N(e) =
+�
+L
+πℏv1
+�
+U0
+2
+�
+m + U0
+2
+�
++ 1
+2
+�
+−
+�
+L
+πℏv1
+�
+U0
+2
+�
+−m + U0
+2
+�
++ 1
+2
+�
++ 1,
+N(o) =
+�
+L
+πℏv1
+�
+U0
+2
+�
+m + U0
+2
+��
+−
+�
+L
+πℏv1
+�
+U0
+2
+�
+−m + U0
+2
+��
++ 1,
+(26)
+with ⌊·⌋ the floor function.
+From the latter, it is clear that at least one even and one odd bound state always exist,
+regardless of the potential width and strength.
+Particularly, for small enough L → 0, one
+obtains the E → 0 and odd E → −m as the even and odd bound state energies, respectively.
+Since the floor function is discontinuous, the number of bound states does not necessarily
+grow continuously for increasing values of L. That is, for L = L0 with N(e,o) bound states, there
+might be a L = L1 > L0 such that (N(e,o) − 1) are generated. This is indeed depicted in Fig. 6a
+for fixed potential depth and different potential length L.
+As discussed in Sec. 2.1, the component ψC might not be continuous, which can lead to
+discontinuous probability densities. Still, one may verify the validity of the bound state eigen-
+values E obtained from (25) by substituting it into (E −U(x))ψC, which should be a continuous
+function3. Particularly, from Fig. 6a, one notices that L = 1 and U0 = 1.5 lead to one even
+3It follows from kyΨB + (U(x) − E)ΨC = 0, which the third of the coupled equations represented by (5).
+11
+
+4
+N(e)
+N(o)
+3
+2
+1
+5
+13
+17
+L0.4
+c
+(E-U(x)Wc
+0
+-0.4
+-2
+-1
+1
+2
+X0.4
+0.2
+0
+-2
+-1
+1
+2
+X(E(e)
+0
+≈ 0.281398) and one odd (E(o)
+0
+≈ −0.32653) bound state energy eigenvalue. The compo-
+nent ψC and (E −U(x))ψC are depicted in Fig. 6b for E ≈ 0.281398, which verifies the required
+continuity condition for the latter function. The same conclusion is drawn for E ≈ −0.32653.
+Furthermore, the corresponding probability distributions associated with �Ψ
+(e) and �Ψ
+(o) are de-
+picted in Fig. 6c in blue-solid and red-dashed, respectively, which are discontinuous.
+5
+Scattering states and transmission amplitudes
+Let us now focus on the scattering of the plane waves on the barrier and the related phenomena.
+This is obtained when plane-wave-like solutions are present in the regions I and III, which
+corresponds to the eigenvalues E ∈ (−∞, − �m)∪( �m, ∞). Without loss of generality, we consider
+only outgoing waves in region III and outgoing together with incoming waves in region I. The
+coefficients of the wave function (16) are then fixed in the following manner,
+αI = 1,
+βI = r,
+αIII = t,
+βIII = 0.
+(27)
+The complex constants t and r can be calculated from (17) as
+t =
+1
+m22
+,
+r = −m21
+m22
+.
+(28)
+The coefficients r and t define the reflection and transmission coefficients R = |r|2 and T = |t|2
+that satisfy R + T = 1. The later expression can be directly verified by substituting from (28)
+when taking into account that there holds m11 = m∗
+22 and m12 = m∗
+21. After some calculations,
+one obtains,
+r = sin(kIIL)
+−2A(B − B′) + i
+�
+A′2 − A2 + (B − B′)2�
+2AA′ cos(kIIL) − i sin(kIIL) ((B − B′)2 + A2 + A′2),
+(29)
+where
+A
+v1
+=
+EkI
+E2 − m2 ,
+A′
+v1
+=
+(E − U0)kII
+(E − U0)2 − m2 ,
+B
+v2
+=
+mky
+E2 − m2 ,
+B′
+v2
+=
+mky
+(E − U0)2 − m2 ,
+(30)
+and kI =
+√
+E2− �m2
+ℏv1
+, kII =
+√
+(E−U0)2− �m2
+ℏv1
+. This expression also holds in cases where solutions in
+the region II are evanescent waves.
+Eq. (29) is a handy expression to understand the transmission of incoming waves from the
+region I and traveling to the region III. Particular interest is paid to cases in which perfect
+tunneling exists, T = 1. Such a tunneling is obtained whenever r = 0, which ensures that t
+is a unimodular complex number. In this case, the incident and transmitted waves share their
+amplitude, but the later carries a relative phase shift t as a leftover of its interaction with the
+barrier. For the sake of clarity, we split our discussion in two cases.
+Normal incidence (ky = 0)
+In this case, the reflection coefficient becomes simpler since B = B′ = 0, kI =
+√
+E2 − m2/ℏv1,
+and kII =
+�
+(E − U0)2 − m2/ℏv1. The numerator in r becomes proportional to m sin(kIIL).
+12
+
+Therefore, for the gapless lattice setup (m = 0), perfect tunneling occurs for any arbitrary ener-
+gies in E ∈ (−∞, −m)∪(m, ∞). This effect was reported in graphene [3,4,43] and pseudospin-1
+lattices [11,44].
+For m ̸= 0, perfect tunneling does exist for specific energies so that kIIL = nπ, with n = 1, . . ..
+The exact resonant energies are straightforward to compute and are presented in a much general
+case below. However, it is worth to analyze the behavior of T = 1 − |r|2 when the barrier is
+large enough, U0 ≫ m, E, for fixed and finite E. The straightforward calculations show that
+T ≈
+1
+1 +
+m4
+4E2(E2−m2) sin2 �
+U0L
+ℏv1
+�.
+(31)
+It reveals that despite the lack of the perfect tunneling, the transmission converges to a non-null
+value as the electrostatic barrier increases indefinitely. This is known as Klein paradox [45], and
+it is in sharp contrast with the non-relativistic case, where transmission becomes smaller for
+larger barrier heights.
+Oblique incidence (ky ̸= 0)
+• Super-Klein tunneling When B = B′ and A = ±A′ in from (29), the reflection coefficient
+vanishes and the transmission becomes perfect (T = 1). This is achieved when E = U0/2. One
+thus has perfect tunneling regardless of the incidence angle for E = U0/2. This phenomenon is
+called the super-Klein tunneling, already reported for pseudospin-1 lattice models with gapless
+dispersion and flat bands [37,44,46], as well as in pseudospin-1/2 graphene lattices [47]. Here,
+we note that the presence of the mass term (m ̸= 0) does not break the super-Klein tunneling as
+long as U0 > 2m. However, super-Klein tunneling is altogether lost by tuning the electrostatic
+barrier such that 0 < U0 < 2m, as no plane-wave solutions exist for E = U0/2. This highlights
+the effects of the mass term (band-gap) on the transmission properties.
+• Generalized Snell-Descartes law It is convenient to define the two-dimensional momentum
+vectors ⃗k = (kI, ky) and ⃗k′ = (kII, ky) that characterize the incident wave and the wave traveling
+through the electric barrier, respectively. The incident and transmitted angles are defined as
+ξ = arctan(ky/kI) and ξ′ = arctan(ky/kII), respectively, see Fig. 7a. Contrary to the bound
+state case of Sec. 4, plane-wave solutions only exist in the region I for bounded values of ky, i.e.,
+|ky| < ky;c =
+√
+E2 − m2/ℏv2. This alternatively implies that scattering phenomenon is available
+for restricted values of the effective-mass term �m. This is depicted in Fig. 7b, from which it is
+also clear that, for |ky| > ky;c, the shaded are covered by the effective-mass region overlaps with
+the energy E, leading to evanescent-wave solutions in the region I.
+From the dispersion relations in the regions I and II, together with the fact that ky is constant
+across all regions, one can establish a relation between the incident and transmitted angles ξ
+and ξ′ of Fig. 7a,
+tan ξ′
+tan ξ
+�
+v2
+1 + v2
+2 tan2 ξ
+v2
+1 + v2
+2 tan2 ξ′ =
+�
+E2 − m2
+(E − U0)2 − m2 .
+(32)
+For v1 = v2, one recovers the same Snell-Descartes law previously reported for graphene [4], and
+to the Snell’s law obtained for pseudospin-1 lattices with m = 0 reported in [46].
+Since we are considering U0 > 2m, we get the following information about the transmitted
+angle:
+13
+
+(a)
+(b)
+(c)
+Figure 7: (b) Scattering configuration (upper-view) for an incident wave ⃗k (region I), with inci-
+dent angle ξ, traveling through an electrostatic barrier (green-shaded area). The wave refracts
+into region II as a wave with vector ⃗k′ and transmitted angle ξ′. (b) Energy configuration of
+the panel (a) with an incident wave with energy E > �m (red-dashed line). (c) Energy curves
+spanned by κ, ky (region I and III) and κ′, ky (region II) for E > �m fixed as in panel (b).
+• For E ∈
+�
+m, U0
+2
+�
+, there exists a transmitted angle ξ′ for every incident angle ξ ∈ (−π/2, π/2).
+• For E = U0
+2 , the transmitted and incident angles are equal, ξ′ = ξ.
+• For E ∈
+� U0
+2 , U0 − m
+�
+∪ (U0 + m, ∞), there are transmitted angles ξ′ ∈ (−π/2, π/2) only
+for ξ ∈ (−ξc, ξc), with the critical angle tan2 ξc = v2
+1
+v2
+2
+(E−U0)2−m2
+2U0
+�
+E− U0
+2
+� . For other values of ξ, the
+solutions in the region II are evanescent waves.
+• For E ∈ (U0 − m, U0 + m), there are only evanescent waves in the region II.
+• Fabry-P´erot resonances Perfect transmission occurs for other energies as well, nevertheless,
+it gets angle dependent. The reflection coefficient (29) vanishes for kIIL = nπ, with n ∈ Z+.
+Since kII is in turn a function of the incidence angle ξ, once may conclude that perfect reflection
+appears only for some specific incidence angles. These are usually known as tunneling resonances
+or Fabry-P´erot resonances [4], and are given as a function of the incident angles ξ as
+E(res)
+±;n =
+�
+1 + v2
+2
+v2
+1
+tan2 ξ
+�
+�
+�
+�U0 ±
+�
+�
+�
+�
+�U 2
+0 −
+1
+1 + v2
+2
+v2
+1 tan2 ξ
+�
+�U 2
+0 − π2v2
+1(n + 1)2
+L2
+−
+m2
+1 + v2
+2
+v2
+1 tan2 ξ
+�
+�
+�
+�
+� ,
+(33)
+with n = 0, 1, . . ..
+These resonant energies behave asymptotically as limξ→±π/2 E(res)
++;n → ∞ and limξ→±π/2 E(res)
+−;n →
+(2U0)−1 �
+U 2
+0 − π2 v2
+1(n+1)2
+L2
+�
+. Thus, for almost perpendicular incident waves (ξ ∼ ±π/2), one re-
+quires larger and larger energies in order to recover the resonances at E(res)
++;n , whereas finite and
+well-defined energy values are required for the resonances E(res)
+−;n . This behavior is depicted in
+Fig. 8a.
+14
+
+I
+I1
+III
+k
+Ki
+k
+3
+1KUo+m
+Uo+m
+Uo
+Uo-m
+Uo-m
+E
+m
+m
+0
+-m
+-m
+II
+IIIRegion
+Region
+1
+II
+K(a)
+Figure 8: (Units of ℏ = 1) (a) Tunneling resonance energies E+;n (blue-solid) and E−;n (orange-
+dashed) as a function of ξ(−π/2, π/2). The inset depicts the transmission amplitude T as a
+function of E for ξ = 0. The shaded area denotes the region where the duple (ξ, E) produces
+evanescent waves in the region I. The parameters have been fixed as v1 = v2 = 1, m = 0.5 and
+U0 = 1.5.
+6
+Remarks on the flat-band solutions
+The piece-wise continuous nature of the electrostatic interaction (10) allows the generation of
+two flat band energies, one located at E = U0 for the region II, and another one at E = 0 for
+the regions I and III. Although the boundary conditions are the same in both cases, the allowed
+matching solutions have a different behavior.
+Let us first consider E = U0 and ky so that plane-wave solutions exist for the regions I and
+III. For generality, we consider incoming and outgoing plane waves in regions I and III, and a
+general flat-band solution in II. Here, the waves entering the interaction zone from the left and
+right have an amplitude I1 and I2, respectively, with I1,2 ∈ R. Additionally, we fix I2
+1 + I2
+2 = 1.
+Under these considerations, we have the general solutions
+Ξ =
+�
+�
+�
+�
+�
+I1ΞkI + A1Ξ−kI
+x < −L/2
+Ξfb
+|x| < L/2
+I2ΞkI + A2Ξ−kI
+x > L/2
+(34)
+where A1,2 ∈ C, Ξ = (mχ, iℏv2kyχ, −iℏv1χ′)T , with χ a complex-valued function, and Ξ±kI the
+solutions (11) evaluated at E = U0. By imposing the boundary conditions (9), one obtains the
+relations A1 = I1e−2iφeikIL and A2 = I2e2iφe−ikIL, with φ = arctan(v2kyU0/mv1kI), whereas the
+arbitrary function χ is restricted to fulfill the following relations at the boundaries,
+χ
+�
+− L
+2
+�
+=
+2v1kII1e−i(φ−kI L)
+�
+m2v1k2
+I + v2k2yU 2
+0
+,
+χ
+� L
+2
+�
+= −
+2v1kII2ei(φ−kI L)
+�
+m2v1k2
+I + v2k2yU 2
+0
+.
+(35)
+Given the arbitrary nature of χ, one may alternatively rewrite it as χ = 2v1kIei(φ−kI L)2x/L
+√
+m2v1k2
+I +v2k2yU2
+0 �χ, where
+�χ(−L/2) = I1 and �χ(L/2) = −I2.
+15
+
+20
+10
+-20
+E
+20
+E(res)
+n
+-10
+-20
+爪
+3
+2
+4
+4
+2Thus, the coupling of incident waves to the flat-band solution leads to a scattering problem
+in which the waves entering the interaction region are completely reflected inside their respective
+regions. Still, the ��at band solutions allowed during such a process must fulfill the boundary
+conditions (35). Note that one also has the conservation property |A1|2 + |A2|2 = I2
+1 + I2
+2 = 1.
+The latter results hold whenever waves enter from only one region, say I1 = 1 and I2 = 0. In
+such a case, we have a perfect reflection in region I, up to a phase in the reflected wave.
+Flat-band solutions also occur for E = 0 in regions I and III. The arbitrary nature of the
+solutions in those flat bands can be tuned so that finite-norm solutions appear. The correspond-
+ing wave function in region II can be found using the boundary conditions, and the calculations
+are as straightforward as the scattering case presented above.
+7
+Concluding remarks
+In this manuscript, it was shown that the existence of a rectangular electrostatic barrier always
+produces at least two bound states for ky = 0, and generates more bound states at different
+energies for increasing values of ky. Interestingly, it was found that even for the asymptotic
+values ℏv2ky → ∞, the associated current density parallel to the barrier is bounded by ±ℏv2,
+where v2 = 2at2. Thus, the current is linear on the hopping amplitude across the ˆy-direction,
+as expected.
+It is worth remarking that dispersion relations obtained from (24) identify the energies for
+which electrons localize in the x-direction, and propagation is allowed in the ˆy-direction is still
+possible. However, by exploiting the separability of free-particle solutions, one can always con-
+struct linear combinations so that electrons localize in the ˆy-direction as well. Such a procedure
+has been discussed in [48] for graphene. For instance, in the example provided in Fig. 5a, one
+can take the energies associated with blue-solid and red-dashed curves as they exist for any
+ky ∈ R. From the relations (26), one can ensure that at least two of such dispersion relations
+always exist. Additional caution must be taken for the other dispersion relations, as they only
+exist for intervals ky ∈ S ⊆ R, and the linear combination must be constructed accordingly to
+that interval. Devising such packages is a task beyond the scope of the current work and will
+be discussed elsewhere, as it deserves attention by itself.
+On the one hand, for the scattering-wave regime, we have proved that even in the gapped
+case (m ̸= 0), the Lieb lattice supports super-Klein tunneling for an energy equal to half of
+the electric barrier, E = U0/2, provided that U0 > 2m. For the gapless case, we recover the
+same results previously reported for gapless T3 lattices [44] and ultra-cold atoms trapped in
+optical lattices [24]. On the other hand, we identified a new modified Snell-like law valid for
+anisotropic Fermi velocities v1 ̸= v2. The latter allows us to identify the Fabry-P´erot resonant
+transmission, which defines a relation between the incident energy and the incident-wave angle
+required to produce perfect tunneling up to a phase factor. Interestingly, for negative energies,
+perfect transmission is achievable for finite energies at incident waves almost perpendicular to
+the barrier. This is not the case for positive energies, as it is shown that the required energies
+diverge.
+The existence of flat-band solutions poses an additional case not available in graphene lattices.
+The latter allows the coupling of the solutions and determining the transmission properties,
+which in this case, leads to perfectly reflected waves. Since the flat-band solutions are defined
+16
+
+in terms of degenerate Bloch waves, there is an infinite family of solutions that allows such a
+reflection, as long as they fulfill the boundary condition (35).
+Acknowledgments
+K.Z. acknowledges the support from the project “Physicists on the move II” (KINE´O II)
+funded by the Ministry of Education, Youth, and Sports of the Czech Republic, Grant No.
+CZ.02.2.69/0.0/0.0/18 053/0017163.
+A
+Determining the number of even and odd bound states
+In this appendix, we present the derivation of the number of bound states for even bound
+states presented in (26). The procedure applies straightforwardly to the odd case as well. It is
+convenient to define the intervals
+I0 =
+�
+0, π
+2
+�
+,
+In =
+�π
+2 + (n − 1)π, π
+2 + nπ
+�
+,
+n = 1, 2, . . . ,
+(A-1)
+so that tan(x) is nonsingular for x ∈ In. Furthermore, if x ∈ ∪k=p2
+k=p1Ik, then tan(x) has (p2 − p1)
+singularities.
+To determine the number of even bound states, one must find the number of interceptions
+of F(E) in (25) and the periodic function tan(kIIL
+2 ) in the interval E ∈ (−m, m). To this end,
+one may notice that F(E) is a monotonously increasing function of E ∈ (−m, m) that tends to
+∓∞ for E → ∓m, and vanishes for E = 0. The latter means that F(E) defines the bijection
+F(E) : (−m, m) �→ R.
+On the other hand, ∂kII/∂E < 0 for E ∈ (−m, m), and one thus
+concludes that tan(kIIL/2) (and also − cot(kIIL/2)) is a monotonously decreasing function of E
+in each of the intervals kIIL
+2
+∈ In, with n = 0, 1, . . .. This property, combined with the fact that
+F(E) : (−m, m) �→ R is a monotonously increasing function, one concludes that interception of
+both functions always exist. One must determine the exact number of interceptions.
+By exploiting the fact that tan(kII L/2) is a periodic function, one just needs to count the
+number of periods inside the interval E ∈ (−m, m) for arbitrary U0 and L, which is equal to the
+number of singularities plus one. m is a lattice parameter, so it is assumed to be a fixed value.
+Be σ± := kIIL
+2 |E=∓m =
+L
+ℏv1
+�
+U0
+2
+�
+±m + U0
+2
+�
+so that the domain of tan
+�
+kIIL
+2
+�
+lies in the interval
+(σ−, σ+) for E ∈ (−m, m).
+Now, if σ− ∈ Ir1 and σ+ ∈ Ir2, with r2 > r1 and r1,2 = 0, 1, . . ., then, tan(kIIL/2) has
+r2 − r1 singularities for E ∈ (−m, m) and intercepts F(E) exactly (r2 − r1 + 1)-times. That is,
+N(e) = r2 − r1 + 1. The values of r1,2 are found by exploiting the fact that ⌊ x
+π + 1
+2⌋ = r for
+x ∈ Ir. One thus has ⌊ σ−
+π + 1
+2⌋ = r1 and ⌊ σ+
+π + 1
+2⌋ = r2, which leads to the expression presented
+in (26).
+The same procedure applies to the odd solutions, where we define the intervals �In = (nπ, (n+
+1)π), with n = 0, 1, . . ., so that cot(x) is nonsingular for x ∈ �In.
+Since − cot(kIIL/2) is
+monotonously decreasing for kIIL/2 ∈ �In, the same same reasoning used in the even case applies
+to the odd case, and one obtains N(o) in (26).
+17
+
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+20
+

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+Application of the partial Dirichlet-Neumann contact algorithm to simulate
+low-velocity impact events on composite structures
+G. Guillameta,∗, A. Quintanas-Corominasa,b,c, M. Riveroa, G. Houzeauxa, M. V´azqueza, A. Turonb
+aBarcelona Supercomputing Center (BSC), Pla¸ca Eusebi G¨uell, 1-3, Barcelona, 08034, Catalonia, Spain
+bAMADE, Universitat de Girona, Av. Universitat de Girona 4, Girona, 17003, Catalonia, Spain
+cDepartment of Civil and Environmental Engineering, Imperial College London, London, SW7 2AZ, UK
+Abstract
+Impact simulations for damage resistance analysis are computationally intensive due to contact algo-
+rithms and advanced damage models. Both methods, which are the main ingredients in an impact event, re-
+quire refined meshes at the contact zone to obtain accurate predictions of the contact force and damage onset
+and propagation through the material. This work presents the application of the partial Dirichlet-Neumann
+contact algorithm to simulate low-velocity impact problems on composite structures using High-Performance
+Computing. This algorithm is devised for parallel finite element codes running on supercomputers, and it
+is extended to explicit time integration schemes to solve impact problems including damage. The proposed
+framework is validated with a standard test for damage resistance on fiber-reinforced polymer matrix com-
+posites. Moreover, the parallel performance of the proposed algorithm has been evaluated in a mesh of 74M
+of elements running with 2400 processors.
+Keywords:
+Contact mechanics, Damage modeling, Finite element analysis, High-Performance Computing
+1. Introduction
+Impacts by foreign objects against any part of the aircraft are a major concern for the aerospace industry
+because they may compromise the structural integrity of the aircraft. Impact events can be classified into
+three main categories: low, high (including ballistics), and hyper-high velocity impacts [1].
+During the
+impact, the energy by the foreign object (projectile) is transferred to the target (structure), and consequently,
+the material can be damaged. Concretely, low-velocity impact events on composite materials (e.g., tool drops
+during maintenance or manufacturing) can drastically reduce the residual strength of the part even for case
+scenarios of Barely Visible Impact Damage (BVID). Therefore, designing composite structures with damage
+resistance cannot be avoided.
+Moreover, extensive experimental campaigns particularly focused on the
+∗Corresponding author
+Email address: [email protected] (G. Guillamet)
+Preprint submitted to Composites Part A: Applied Science and Manufacturing
+January 16, 2023
+arXiv:2301.05552v1  [cs.CE]  11 Jan 2023
+
+investigation and evaluation of damage resistance of a specific material may be prohibitive by the industry
+in terms of costs.
+Thus, virtual testing of impact events is of great interest as mathematical models and technology advance.
+However, solving the physics behind this problem, particularly from the material point of view, is still one
+of the most complex and challenging problems today. A review of existing software for composite impact
+modeling focused on low-velocity events is conducted by Nguyen et al. [38]. In this review, the constitutive
+damage models play an essential role apart from the methods such as the contact algorithm or temporal
+integration scheme. Most of them can capture the trends and peak forces reasonably well. The research
+community has put and continues to put a lot of effort into developing reliable constitutive damage models
+for composites. Remarkable progress has been made on specific methodologies and constitutive damage
+models for predicting the damage resistance and damage tolerance of composite structures [28, 10, 17, 53,
+29, 52, 51, 15]. However, in terms of computational performance, the resolution of an impact problem,
+including different sources of damage, is still computationally demanding. The use of sophisticated contact
+algorithms and advanced damage models require refined element meshes to accurately predict the onset and
+propagation of the damage in such materials.
+The most commonly used contact algorithms for the resolution of an impact problem are the Penalty
+methods [21], Classical Lagrange multipliers [4, 16] or the Augmented Lagrange multipliers. The latter is
+often chosen to solve the contact inequality constraints, see [58]. However, the parallel aspects of these
+traditional contact algorithms are not trivial, and to the authors’ knowledge, little effort has been invested
+in the parallel aspects of such algorithms and their scalability in supercomputers. Some research works
+dealing with the parallel aspects of contact algorithms are [34, 35, 21].
+A completely different approach to the previous algorithms is the method of partial Dirichlet-Neumann
+(PDN) conditions. The contact is tackled as a coupled problem, in which the contacting bodies are treated
+separately, in a staggered way. The coupling is performed through the exchange of boundary conditions at the
+contact interface following a Gauss-Seidel strategy. The pioneering works using this approach are conducted
+by Krause and Wohlmuth [24] and Yastrebov [59], showing the capabilities of solving nonlinear contact
+problems. To the authors’ knowledge, one of the first applications of this method for explicit dynamics is
+made by Lapeer et al. [25], where the PDN method was used to simulate natural childbirth using explicit
+dynamics and executed in a hybrid system with Central Processing Units (CPU) and Graphics Processing
+Unit (GPU) architectures. However, little attention is dedicated to the computational performance and
+the parallel aspects of dealing with large-scale models. More recently, this method has been adapted and
+implemented in parallel in the Alya multiphysics code [57] by Rivero [46] and published by the authors in
+[19]. The mathematical and the parallel aspects are described in detail in these works, demonstrating the
+benefits of the PDN contact algorithm in High-Performance Computing (HPC) systems.
+In this paper, we present the application of the aforementioned method proposed by the authors in [46, 19]
+2
+
+for the resolution of low-velocity impact problems for composite materials. Existing time integration schemes
+and constitutive models from the literature have also been adapted and implemented within a parallel
+framework. So the main contribution of the present paper is focused on the extension of the PDN contact
+algorithm for explicit time integration schemes and its use in HPC systems involving impact events in the
+field of composite materials. Additionally, a new mesh multiplication algorithm is presented to deal with
+cohesive elements and element technologies such as continuum shell elements.
+The content of this paper is structured as follows. Firstly, the methods for the resolution of low-velocity
+impact events on composite materials including damage are explained with a strong emphasis on the contact
+algorithm and its implementation in parallel codes based on the finite element method. Then, the algorithm
+is validated through three benchmark tests.
+The first one consists of a quasi-static indentation test to
+verify that the contact pressure is well captured by using implicit and explicit time integration schemes.
+The second and the third examples correspond to a low-velocity impact on a composite plate following the
+ASTM International standard to measure the damage resistance of fiber-reinforced polymers. These last
+examples, use different material systems which are quite used by the aerospace industry, the T800S/M21 and
+the AS4/8552 carbon/epoxy systems. The numerical predictions obtained are correlated with experiments,
+and the computational performance is analyzed and discussed for the coupon made of AS4/8552 material.
+Finally, the conclusions of this work are commented on together with future work to improve the simulation
+of impact events including damage.
+2. Modeling framework for low velocity impact events using High-performance systems
+This section describes the modeling framework and the application of the partial Dirichlet-Neumann
+(PDN) contact algorithm for the simulation of impact events. Particular emphasis is put on the extension of
+such contact algorithm for explicit dynamic analysis. All the methods presented here are implemented in the
+Alya multiphysics code [57] based on the Finite Element Method (FEM). This parallel code is based on high-
+performance programming techniques for distributed and shared memory supercomputers. Moreover, the
+methods are programmed using the total Langrangian formulation, where stresses and strains are measured
+with respect to the original configuration. The Green strain measure and the 2nd Piola-Kirchoff stress are
+used, and we follow the notation from Belytschko et al. [6] throughout the paper. As an impact event is
+a complex and computationally demanding engineering problem is very attractive to be solved using High-
+Performance Computing. It is worth highlighting that all the methods described here can be implemented
+in other parallel FEM codes.
+2.1. Partial Dirichlet-Neumann contact algorithm
+The low-velocity impact event proposed in this paper can be assumed as a non-linear contact problem,
+where the striker is considered as a rigid body and the plate as a deformable body. Let’s assume that both
+3
+
+body instances are of arbitrary shape, and we do not consider friction. Therefore, this contact problem can
+be written as a boundary value problem, see Yastrebov [60], which includes the Hertz-Signorini-Moreau law
+for normal contact. So the balance of momentum and the contact conditions can be written as follows:
+∇ · σ + f v = 0
+in Ω
+σ · n = σ0
+on ΓN
+u = u0
+on ΓD
+g ≥ 0, σn ≤ 0, σn g = 0, σt = 0
+on ΓC
+(1)
+being σ the Cauchy stress tensor, f v a vector of volumetric forces, σ0 a set of prescribed tractions on the
+Neumann boundary, ΓN; u0 a set of prescribed displacements on the Dirichlet boundary, ΓD. Over the
+contact boundary, ΓC, we have imposed the following conditions: g represents the gap between contacting
+bodies, σn is the normal contact pressure, and σt is the tangential stress. The tangential stress equal to
+zero (σt = 0) in Eq. (1) characterizes a frictionless contact case.
+In order to satisfy the conditions in Eq. (1), the present paper uses the partial Dirichlet-Neumann
+contact algorithm proposed by Rivero [46, 19] which is based on the work from Yastrebov [60]. In the works
+mentioned above, the method was applied for implicit time integration schemes, while in the present work,
+the algorithm is extended to explicit schemes. The main benefits of the PDN contact algorithm to typical
+Penalty or Lagrange Multipliers methods are the following: (i) the size of the problem does not increase due
+to the Lagrange multipliers methods as unknowns (ii) no restriction with respect to the mesh partitioner
+due to the use of contact elements (iii) absence of contact tangent matrices (implicit schemes) and residual
+contact force vectors and (iv) easy to be parallelized as it can be treated as a solid-to-solid coupling using
+existing methods for multiphysics applications such as the Gauss-Seidel scheme.
+The iterative process of the PDN contact algorithm in a frictionless problem is shown in Fig. 1. Let’s
+assume that the time of the simulation is 0 < t < tE and it is subdivided into nT S time steps ranging
+from n = 1...nT S and tE is the time at the end of the simulation. At time step n there is no interaction
+between both code instances, so no contact is detected (Fig. 1a). Then, at time tn+1, contact is detected
+as we have overlapping between both bodies. At this step, the non-penetration boundary conditions are
+treated kinematically, i.e., as Mulitple Point Constraints (MPC) by the projection of the nodes belonging
+to the slave surface (deformable’s body) to the master surface (rigid’s body) using a Dirichlet condition.
+Then, a local coordinate system with normal-tangent basis vectors nj and tj is created for each detected
+node j. The contact node is restricted to only move to the tangent line defined by the vector tj, see Fig.
+1e. In a hypothetical frictional contact problem, the friction force would be imposed in this direction as a
+Neumann boundary condition. In this work, friction is not considered for the low-velocity impact as the
+relative velocities at the contact zone are sufficiently small. After that, the contact algorithm checks the
+4
+
+2
+(a)
+(c)
+(b)
+(d)
+Rigid 
+Deformable
+Contact node
+Released (free) node
+(e) Kinematic constraint for node 1 and 2
+n2
+t2
+n1
+t1
+1
+2
+f c
+f c
+f c
+f c
+f c
+f c
+Figure 1: Iterative process of the parallel PDN contact algorithm. (a) Interaction; (b) Overlapping; (c) Dirichlet boundary
+conditions (projections) (d) Released nodes and equilibrium. (e) Kinematic constraint for node 1 and 2. The reader is referred
+to the web version of this paper for the color representation of this figure.
+presence of adhesion or artificial contact nodes, i.e., nodes in traction (Fig. 1c). The reaction contact force
+f c
+j has to satisfy the following condition:
+f c
+j · nj ≥ 0
+(2)
+Those adhesion nodes have to be released, so the current time step tn+1 has to be repeated; the i index
+shown in Fig. 1c and 1d represents the sub-iterations for node release. The whole kinematic constraint
+process is depicted for two of the contacting nodes in Fig. 1e. The nodes release algorithm for explicit time
+schemes is described in Algo. 2 in Appendix
+A. The condition to distinguish a true contact node or an
+adhesion (artificial) contact node is by means of the contact force (reaction due to the Dirichlet condition).
+The vector of contact forces using total Lagrangian formulation can be expressed as:
+(f c
+j)T =
+�
+Ω0
+BT
+0jP dΩ0
+(3)
+where BT
+0j is the matrix containing the derivatives of the shape functions with respect to the reference
+system and P is the nominal stress tensor, see [6].
+An exciting aspect of the PDN method is that the computational cost of the projections is very small
+compared to Penalty or Langrange approaches [25]. One of the most consuming parts and a vital issue for
+further research is the contact searching and communication between the subdomains (belonging to different
+code instances), as stated in the previous work from the authors [19]. In our case, we use the PLE++ library
+[61], which is an adaptation of the Parallel Location and Exchange PLE library [14]. The main algorithm
+of the PDN contact method is described in Algo. 1 in Appendix
+A and the nodes release algorithm for
+5
+
+explicit schemes is summarized in Algo. 2. The reader is referred to Rivero [46] and Guillamet et al. [19]
+works for more details on the implementation aspects.
+2.2. Time integration schemes
+2.2.1. Deformable body
+Spurious oscillations may appear when using explicit time schemes for dynamic and wave propagation
+problems such as impact events. These oscillations occur due to the mismatch of two different types of wave
+components. Thus, dissipative explicit time schemes are often used to reduce the numerical instabilities
+induced by the spatial and time discretization procedures. Among the many dissipative methods available,
+the Tchamwa–Wielgosz (TW) explicit scheme [31] is beneficial because it damps out the spurious oscillations
+occurring in the highest frequency domain. This is the time integration scheme selected in this work, but
+any other explicit time scheme such as the Central Difference (CD) [6] including bulk viscosity could also
+be used.
+The motion described by the TW scheme is the following:
+˙d
+n+1 = ˙d
+n + ∆t¨d
+n
+(4)
+dn+1 = dn + ∆t ˙d
+n + ϕ(∆t)2¨d
+n
+(5)
+where d, ˙d, ¨d are the displacement, velocity and acceleration nodal vectors, respectively; ∆t is the time
+increment or step size; and ϕ is a numerical viscous parameter, which in the current work is set to 1.033
+[31]. The key to the computational efficiency of explicit time integration schemes is the use of the lumped
+mass matrix for the resolution of the linear system of equations, which is simplified as an easy inversion of
+the diagonal mass matrix [6]. The global stiffness matrix is not required to be assembled as it is needed for
+implicit time integration schemes. The explicit time integration scheme solves accelerations, so their values
+at the beginning of the increment are computed by making use of the equation of motion of the system:
+m ¨d
+n = f n(dn, tn) = f e(dn, tn) − f i(dn, tn) − f c(dn, tn)
+(6)
+where m is the vector representation of the lumped mass matrix, f e is the global vector of external forces, f i
+is the global vector of the internal forces, and f c is the global vector of the contact forces from the Dirichlet
+condition imposed on that nodes. Thanks to m, the acceleration can be computed without invoking any
+solver as:
+¨d
+n = m−1(f e − f i − f c)
+(7)
+6
+
+2.2.2. Rigid body
+The striker in the present work is considered as a rigid body. The resolution of the equations of motion
+for the rigid bodies we use a 4th order Runge Kutta scheme. Let’s consider the following differential equation
+where the right hand side is a function of both time and another function dependent on time.
+dy
+dt = f(t, y(t))
+(8)
+From this equation, the Runge-Kutta method estimates the solution at n + 1 taking into account four
+evaluations of the right hand side step dt as follows,
+k1 = dt · f(t, y(t))
+k2 = dt · f(t + dt
+2 , y(t) + k1
+2 )
+k3 = dt · f(t + dt
+2 , y(t) + k2
+2 )
+k4 = dt · f(t + dt, y(t) + k3)
+yn+1 = y(t + dt) = y(t) + k1
+6 + k2
+3 + k3
+4 + k4
+6
+(9)
+In the present paper, the motion of the striker is solved by making use of the following differential
+equation:
+m · ¨d = m · g − f e
+(10)
+where m is a scalar value of the mass of the rigid body, g is the gravity force vector at the center of mass,
+¨d is the linear acceleration, and f e is the external force also at the center of mass from the rigid body. It is
+worth mentioning that the rigid body is represented by a point (center or mass), so the above vectors have
+a dimension of 2 for 2-d problems and 3 for 3-d problems. When contact occurs the external force from the
+rigid body is calculated by f e = �nc
+j=1 f c
+j, where j denotes a contact node and nc is the total contact nodes
+belonging to the deformable body.
+2.3. Mesoscale damage modeling for fiber-reinforced composites
+The mesoscopic length scale is the most suitable for virtual testing of low-velocity impacts on structures
+made of composite materials. At this scale, the numerical predictions have a good trade-off between infor-
+mation about the damage mechanisms driving the failure process and the structural response without the
+complexity of dealing with intricate microstructures. It is worth emphasizing that the mesoscopic length
+scale is not only appropriate for the bottom levels of the building block approach (coupon and elements)
+[28, 53, 15, 50] but also for the top levels (sub-components and components) [43, 44, 13].
+7
+
+From the constitutive modelling viewpoint, the mesoscopic length scale simplifies the intricate microstruc-
+ture of long fibre composite laminates by homogenising the properties and mechanisms at the lamina level.
+The outcome is a layered material with two well-defined regions: intralaminar and interlaminar. The former
+is modelled as a transversally isotropic material, which can fail due to fibre breaking and matrix cracking
+according to the loading scenario. The latter is modelled as a very thin region, usually tending to zero
+thickness, where delamination can onset and propagate.
+Regarding the modelling architecture, several strategies exist in the literature suitable for modelling
+composite at a mesoscopic length scale using FEM [37]. We adopt a continuous approach for the intralam-
+inar region (CDM with linear elements) and a discontinuous one for the interlaminar (CZM with interface
+elements). The straightforward implementation of this strategy in a standard FEM code aids in preserv-
+ing the scalability of Alya multiphysics [41]. Thus, the mesoscale damage modeling strategy exploits the
+computational resources to maximize the accuracy of the impacts thanks to very thin meshes.
+2.3.1. Intralaminar damage model
+The intralaminar damage model for predicting ply failure is based on the continuum damage mechanics
+framework.
+Fiber and matrix cracks are smeared in the continuum and represented by state variables.
+Accordingly, the crack’s kinematics is not explicitly represented, but their effects on the degradation of the
+capacities of sustaining loads. In turn, the onset and growth of the damage failure mechanisms are governed
+by the failure surfaces and evolution laws. In this work, we employ a local damage model based on the
+constitutive modeling framework for long fiber composite materials proposed by Maim´ı et al. [32, 33]. This
+framework has been used widely in the literature, demonstrating outstanding accuracy and performance not
+only for static scenarios [11, 7, 41] but also for impact [17, 51, 48] and fatigue [26, 27].
+The main ingredients of the intralaminar damage model are: i) transversally isotropic elasto-plastic re-
+sponse, ii) damage activation functions related to the different ply failure mechanisms through the maximum
+strain criterion for the fiber breaking and the LaRC criteria for the matrix cracking, iii) the damage evolution
+laws are defined to dissipate the fracture energy associated to the opening mode ensuring mesh objectivity
+by the crack-band theory [5], and iv) the thermodynamic consistency is ensured by imposing irreversibility
+of the damage variables.
+Fig. 2 illustrates the intralaminar failure modes schematically modelled, while Algo. 4 in Appendix A
+summarises the material model workflow. Note that a plastic response under shear loads is considered,
+and five damage mechanisms are modelled: fibre breaking, fibre kinking, tensile and compressive matrix
+cracking, and shear matrix cracking. The details of the expressions employed and their justification from a
+physical standpoint can be found in [32, 33, 51].
+Besides the constitutive response, the intralaminar damage model also encloses the computation of the
+critical time step, which is required by the explicit time integration scheme. For the sake of simplicity, we
+8
+
+Fibre breaking
+Fibre kinking
+Matrix cracking
+Matrix cracking
+Matrix cracking
+Figure 2: Schematic representation of the intralaminar damage mechanisms. Adapted from [26].
+utilise the same formula of a transversally isotropic material:
+∆t = vsound
+ℓc
+=
+�
+max Cij
+ρ
+(11)
+where Cij are the components of the effective stiffness matrix, ρ is the density of the material, and ℓc is
+the characteristic element length. Considering a structured hexahedral mesh employed, we approximate the
+characteristic element length with the element volume Ve [33]:
+ℓc ≈
+3�
+Ve
+(12)
+2.3.2. Interlaminar damage model
+The interlaminar damage model for predicting the onset and propagation of delamination is based on
+the cohesive zone approach and formulated in the context of damage mechanics. Accordingly, a damage
+state variable is employed to account for the gradual loss of the bearing capacities of the material in the
+cohesive zone due to the separation of crack surfaces. In turn, the separation or opening of the crack is
+represented by a kinematic quantity noted as displacement jump, which is approximated by employing the
+interface element technology [2, 39, 9]. Thus, the interlaminar damage model is a constitutive model that
+computes the cohesive reactions as a function of displacement jumps. More details of the mesoscale modeling
+of delamination using cohesive zone models can be found in Carreras et al. [12].
+In this work, we use the cohesive zone model proposed by Turon et al. [54, 56], which has been employed
+extensively in the literature [20, 47, 40, 42]. The main characteristics of this CZ model are: i) linear response
+9
+
+125 μm125 μm125 μm125 μm125 μmbefore initiation of the softening, ii) linear relation between the cohesive tractions and crack openings,
+iii) onset and propagation of the damage in compliance with the Benzeggagh-Kenane criterion, and iv)
+thermodynamic consistency despite the loading scenario, even when the mix-mode ratio varies. Algo. 5
+summarizes the cohesive zone model workflow.
+Regarding the element technology, we employ zero-thickness interface elements for capturing the delam-
+ination, implemented using the formulation presented in [45]. As standard interface elements are used, the
+integrals are computed using a Newton-Cotes integration scheme to mitigate the spurious oscillations in the
+traction profile along the interface [49]. The stable time increment, which is necessary for the explicit time
+integration scheme is obtained through [51]:
+∆tcoh =
+�
+¯ρ
+Kcoh
+(13)
+where ¯ρ and Kcoh are numerical parameters known as cohesive surface density and penalty stiffness, re-
+spectively. The cohesive surface density for zero-thickness elements is approximated by the expression in
+[52]. In turn, the cohesive penalty stiffness is defined to avoid affecting the compliance of the system as
+Kcoh ≥ 50ET /tlam, where ET is the transverse elastic modulus and tlam the adjacent laminate thickness
+[55].
+2.4. Mesh refinement algorithm for interface elements with a cohesive law
+The mesh multiplication algorithm proposed by Houzeaux et al. [22] has been extended to deal with
+the presence of interface elements or even continuum shell element formulations.
+Focusing on interface
+elements, they are zero-thickness elements with a cohesive material law that are inserted between plies in
+a laminated composite material in order to predict the delamination damage mechanism. It is well known
+that an accurate prediction of the onset and propagation of delamination in composite materials requires
+very refined meshes, as stated in [54, 55]. However, depending on the number of interface layers or the
+geometry size, it can be challenging to place these elements between plies and computationally demanding
+to solve the problem.
+On the other hand, mesh generation of large meshes is often a bottleneck in engineering applications
+to deal with thousands of millions of elements. Thus, integrated tools for mesh refinement within parallel
+codes devised for High-Performance systems allow a parallel and fast refinement of the coarse mesh without
+the need to create the mesh again.
+Therefore, this paper also introduces a new capability of the mesh multiplication algorithm from [22],
+which enables the refinement of large-scale problems, including interface elements. Let’s assume a configu-
+ration of two bulk elements together with an interface element between them, as shown in Fig. 3.
+10
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+Interface
+element 
+Solid
+element 
+2
+3
+4
+1
+6
+7
+8
+5
+Original node
+New edge node
+New face node
+New center node
+2
+3
+4
+6
+8
+5
+(ne
+ELINT= 4) 
+Bulk element 
+Interface element 
+(ne
+BULK = 8) 
+Global numbering
+Local numbering
+1
+7
+Figure 3: Mesh multiplication between bulk and interface (cohesive) elements.
+The 8-node interface element can only be divided into four elements to avoid the duplication of the
+element at the interface mid-plane between the bulk elements. The criterion used for the correct division
+is by making use of the element normal, also known as stacking direction, which is required for the proper
+behaviour of the element due to its kinematics. Thus, those parallel planes to the element normal are used
+to divide the element. The dimensions of the new mesh can be calculated as follows:
+ne = 8 · n0
+e,BULK − n0
+e,ELINT · 4
+nn = n0
+n + nedges + nfaces + n0
+e,BULK − n0
+e,ELINT − n0
+edges,ELINT − n0
+faces,ELINT
+nb = 4 · n0
+b − 2 · n0
+b,ELINT
+(14)
+where ne, nn and nb are the total number of elements, nodes and boundaries for the new mesh. In order
+to refine the hybrid mesh is important to know the total number of n0
+e, n0
+n and n0
+b from the original mesh
+and also information about the edges and faces that have to be divide or not. Algo. 3 summarizes the
+different steps and functions for the mesh division and reconstruction of the interface domains in a parallel
+framework.
+3. Benchmark tests
+Three benchmark tests are conducted to validate the application of the parallel partial Dirichlet-Neumann
+contact algorithm using an explicit time integration scheme. The first example consists of a quasi-static
+indentation test, which has already been solved using an implicit time integration scheme in [19].
+The
+solution using explicit analysis is compared with the numerical solution obtained for implicit analysis. The
+second and the third examples consist of a low-velocity impact event on two coupons manufactured with two
+11
+
+well-known material systems for the damage prediction: T800S/M21 and AS4/8552 respectively. Thanks
+to the proposed algorithm’s flexibility and generality, we use a multi-code approach, where the motion of
+each body (rigid and deformable) is solved using different instances of Alya. Regarding the partitioning
+of the mesh, we use the Space-Filling Curve (SFC) based partitioner described in [8], which performs the
+partitioning in parallel and maximizes the load balance. It is worth highlighting that all the executions
+here are in parallel (pre-process, solution, and post-process steps). In all the examples, the contact bodies
+are discretized with a refined finite element mesh to assess the geometrical localization between both code
+instances and to obtain an accurate prediction of the contact force. All the simulations are conducted in
+MareNostrum4 supercomputer. This cluster has 3456 nodes, each of them with 48 processors Intel Xeon
+Platinum @ 2.1 [GHz], giving a total processor count of 165 888 processors.
+3.1. Quasi-static indentation test
+This example has already been solved using an implicit time solution scheme in [19, 46]. This case is
+now solved as a quasi-static problem using explicit dynamics. The example consists of a rigid rounded head
+(indenter) and a deformable beam, see Fig. 4. The geometrical dimensions of the indenter (rigid body) are
+ri = 1 m and wi = 0.5 m, while the beam (deformable body) are hb = 0.25 m, lb = 1.5 m and wb = 0.3 m. The
+relative position of the indenter with respect to the beam is given by the parameters ax = 0.25 m, az = 0.1 m
+and ay = 0.01 m (gap). The beam is modelled with an hyperelastic Neo-Hookean formulation [3] and finite
+strains, with material properties Eb = 6.896 × 108 Pa (Young modulus), νb = 0.32 (Poisson ratio) and density
+ρ = 1000 kg m−3. The beam is fully clamped at the bottom face, and a prescribed vertical displacement of
+δ = 0.11 m is applied at the top surface belonging to the indenter. Both bodies are discretized with finite
+elements using full integration: 8-node linear solid elements for the beam and 4-node linear tetrahedrons
+for the indenter. The beam has a base mesh of 3510 elements, while the indenter has 15 960 elements. A
+non-linear dynamic analysis is performed with a total time of the simulation of 0.05 s and a fixed time step of
+1 × 10−5. The selected time step value is smaller than the stable time increment, which is 2.796 × 10−5, and
+no mass scaling is used. In order to perform a quasi-static event and minimize the kinetic energy, a smooth
+step function (fifth-order polynomial) is applied. This function has the form A0+(AE −A0)ξ3(10−15ξ+6ξ2)
+for t0 ≤ t < tE, where A0 and A1 are the initial and final amplitude, t0 and tE are the initial and final time
+of the simulation and ξ =
+t−t0
+tE−t0 . This smooth load rate ensures that the first and second time derivatives
+are zero at the beginning and the end of the transition.
+12
+
+Beam (deformable)
+hb
+x
+y
+z
+y
+lb
+wi
+wb
+az
+ax
+Indenter (rigid)
+ri
+ux = uy = uz = 0
+ux = uz = 0,
+ay
+uy =
+Figure 4: Setup for the quasi-static indentation test. Adapted from [19].
+Displacements and forces obtained at the contact zone are shown in Fig. 5 for two different paths. Line
+path a is centered and goes from one side to the other in the length direction of the beam, while line path b
+is also centered in the width direction. The numerical prediction using the explicit time integration scheme
+is compared with the implicit solution obtained in [19]. We can observe an excellent agreement between
+both numerical predictions in terms of the displacements and contact force.
+(a)
+(b)
+(c)
+(e)
+(d)
+Path b
+Path a
+[18]
+[18]
+[18]
+[18]
+[18]
+Figure 5: Displacements and contact forces at straight lines path a and b. (a) Tangential displacement in x-direction for path
+a. (b) Normal displacement in y-direction for path a. (c) Tangential displacement in z-direction for path b. (d) Contact force
+at line path a. (e) Contact force at line path b.
+13
+
+3.2. Low velocity impact on a composite plate
+The proposed benchmark consists of a drop-weight of a rigid hemispherical striker on a rectangular plate
+made of composite material, see Fig. 6. Two impact scenarios using different material systems, layups, and
+impact energies are considered for the validation of the proposed framework. The materials selected are
+the unidirectional prepreg M21/194/34%/T800S (T800S/M21) and the unidirectional prepreg AS4/8552,
+both carbon-epoxy systems. On the one hand, the coupon made of T800S/M21 is manufactured by Hellenic
+Aerospace Industry and tested at Element Materials Technology Seville facilities within the framework of the
+CleanSky2 SHERLOC project. Most of the material properties from the T800S/M21 are also characterized
+by Hellenic Aerospace Industry and Element Materials Technology Seville. On the other hand, the coupon
+made of AS4/8552 is chosen from literature through the works conducted by Gonz´alez et al.
+[17] and
+Soto et al. [51]. All the material properties from the aforementioned materials, including damage model
+parameters, are summarized in Tab. 1. The intralaminar damage model is fed by the in-situ strengths which
+are calculated following the works by Furtado et al. [15] and Soto et al. [51].
+Rubber clamp
+(ux = uy = uz = 0)
+Striker
+rs = 8 mm
+(ux = uy = 0)
+ms = 5 kg
+Top view
+ 
+75 mm
+125 mm
+R7
+Refined region
+(75 mm x 75 mm)
+ 
+(uz = 0)
+Window cut
+ 
+Stacking sequence
+ [454/04/-454/904]s
+Cohesive elements between clusters
+tply = 0.18125 mm
+tcoh = 1.0 x 10-4 mm
+Figure 6: Numerical setup for the low velocity impact test. The mesh and layup correspond to the coupon made of AS4/8552
+material used for the parallel performance analysis.
+Both impact case scenarios follow the standard ASTM D7136/D7136M-20 [23] for damage resistance
+evaluation of fiber-reinforced polymers. Each plate has the same dimensions: 150 mm×100 mm and each
+of them are supported on a metallic frame with a cut-out of 125 mm×75 mm.
+Rubber-tipped clamps
+clamp the plate instance at the four corners. We consider equivalent boundary conditions to represent this
+experiment. As we can see in Fig. 6, the metallic frame and the rubber clamps from the experiment do not
+exist as physical entities, so we only consider the contact surface from the rubber cylinder-shaped clamps
+and the contact edges of the cut-out window of the metallic frame, where we apply the boundary conditions.
+14
+
+T800S/M21
+AS4/8552
+Property
+Value
+CV(%)
+Ref.
+Value
+Ref.
+Density (t/mm3)
+1.59 × 10−9
+-
+1.59 × 10−9
+[51, 17]
+Elastic
+E11 (MPa)
+138.4 × 103
+1.95
+128.0 × 103
+[51, 17]
+E22 = E33 (MPa)
+8.54 × 103
+3
+7.63 × 103
+[51, 17]
+ν12 = ν13 (-)
+0.311
+16
+0.35
+[51, 17]
+ν23 (-)
+0.45
+-
+0.45
+[51, 17]
+G12 = G13 (MPa)
+4.29 × 103
+3
+4.358 × 103
+[51, 17]
+G23 (MPa)
+2.945 × 103
+-
+2.631 × 103
+[51, 17]
+Strength
+XT (MPa)
+2854.0
+4
+2300.0
+[51, 17]
+XC (MPa)
+1109.0
+13
+1531.0
+[51, 17]
+YT (MPa)
+56.6
+5.8
+74.2
+YC (MPa)
+250.0
+[15]
+199.8
+[51, 17]
+SL (MPa)
+93.7
+0.6
+94.36a
+[36]
+αo (◦)
+53
+[32]
+53
+[32]
+In-situ strengthsc
+Y is
+T,int (MPa)
+132.5 (1tply)
+-
+117.5 (4tply)
+Y is
+T,int (MPa)
+93.7 (2tply)
+-
+117.5 (8tply)
+Y is
+T,out (MPa)
+83.8 (1tply)
+-
+74.2 (4tply)
+Y is
+C,int (MPa)
+250.0 (1tply)
+-
+199.8 (4tply)
+Y is
+C,int (MPa)
+250.0 (2tply)
+-
+199.8 (8tply)
+Y is
+C,out (MPa)
+250.0 (1tply)
+-
+199.8 (4tply)
+Sis
+L,int (MPa)
+116.0 (1tply)
+-
+120.8 (4tply)
+Sis
+L,int (MPa)
+116.0 (2tply)
+-
+120.8 (8tply)
+Sis
+L,out (MPa)
+93.7 (1tply)
+-
+94.4 (4tply)
+Fracture toughness
+GXT (N/mm)
+340
+[15]
+81.5
+[51, 17]
+GXC (N/mm)
+60.0
+[15]
+106.3
+[51, 17]
+GY T (N/mm)
+GIc
+7.3
+GIc
+[51, 17]
+GY C (N/mm)
+1.38b
+20
+1.313b
+[51, 17]
+GSL (N/mm)
+GIIc
+20
+GIIc
+[51, 17]
+Traction separation law
+fXT (-)
+0.1
+-
+0.1
+[51, 17]
+fGT (-)
+0.6
+-
+0.6
+[51, 17]
+fXC (-)
+0.1
+-
+0.1
+[51, 17]
+fGC (-)
+0.9
+-
+0.9
+[51, 17]
+Matrix plasticity
+Sp (N/mm)
+66.9
+-
+[15]
+62.0a
+Kp (N/mm)
+0.09
+-
+[15]
+0.1936a
+Interface properties
+GIc (N/mm)
+0.308
+7.3
+0.28
+[51, 17]
+GIIc (N/mm)
+0.828
+20
+0.79
+[51, 17]
+τI (MPa)
+49.2d
+5.8
+YT
+[51, 17]
+τII (MPa)
+80.7d
+0.6
+SL
+[51, 17]
+η (-)
+1.75
+-
+1.45
+[51, 17]
+Kcoh (M/mm3)
+1.1 × 106
+-
+2.5 × 104
+[51]
+a Best fitted based on properties from [36]
+b GY C = GSL/cos(αo) [32]
+c Calculated considering plasticity using equations from [51]
+d Engineering solution by Turon et al. 2007 [55] using Ne=5
+Table 1: Material properties for the M21/194/34%/T800S (T800S/M21) and Hexply AS4/8552 including damage models
+parameters.
+15
+
+The velocity of the striker is given as an initial condition set in the impact direction, while the remaining
+degrees of freedom are constrained. The initial velocity of the striker is calculated based on the impact
+energy of each case study. The initial position of the striker has a gap of 0.01 mm between the striker
+tip and the top surface of the plate in order to avoid overlapping between bodies at the beginning of the
+simulation. Moreover, gravity forces are included in both body instances, considering a gravity value of
+9.81 m/s2.
+We employ 8-node full integration hexahedron elements for the plate using the inter- and intra-laminar
+damage models described in Sec. 2.3.2 and Sec. 2.3.1 respectively. Cohesive elements are inserted at each
+interface between different ply angles.
+It is worth highlighting that other constitutive material models
+and element technologies would also be feasible in combination with the proposed contact algorithm. With
+regards to the strikers used for each impact case scenario, they are discretized with 4-node linear tetrahedron
+elements with a biased mesh of 0.1 mm at the center of the half-sphere and 1 mm at the end of the edge.
+The total number of elements for the striker used for the T800S/M21 and AS4/8552 materials are 32 685
+and 79 934, respectively. Regarding the plates, they both have a refined centered region of 75 mm×75 mm
+with an in-plane element size equal or multiple to the ply thickness, depending on the material system in
+order to guarantee an aspect ratio close or equal to 1.
+3.2.1. Coupon made of T800S/M21 material
+This impact coupon has a stacking sequence of [45/ − 45/02/90/0]S and is made of T800S/M21. The
+nominal ply thickness is 0.192 mm. This case study is submitted to an impact energy of 10 J, which falls into
+the Barely Visible Impact Damage (BVID) analysis. The striker has a diameter of 25 mm and a mass of 2 kg,
+which is modeled as a rigid body. The global element size for the plate is 1 mm, and each lamina and the
+clusters of two plies have one element through the thickness. The in-plane element size is 0.192 mm which
+is equal to the ply thickness resulting in an aspect ratio of 1 for those elements at plies without clustering
+and located at the refined region. The mesh of the plate has a total of 1 042 525 hexahedron elements (≈
+3.3 million of Degrees Of Freedom (DOF)). The total time for this simulation is set to 5.0 ms. The initial
+velocity of the striker considering the gap previously mentioned is 3.16 m/s.
+The numerical predictions for this impact case scenario and their comparison with experimental data
+are shown in Fig. 7 and summarized in Tab. 2. The experimental test campaign consisted of testing a
+batch of five coupons to ensure proper repeatability of the results. The force-time for each impact was
+recorded with a limited number of points (52 points on average for each impact test). The reduced number
+of points only allows for validation of the global behavior of the impact case scenario. Energy-time and the
+force-displacement curves are calculated by integrating once and twice the experimental force history curve.
+16
+
+(a)
+(b)
+(c)
+Figure 7: Experimental and numerical curves for the 10J impact on the coupon made of T800S/M21 material. (a) Impact
+force-time. (b) Impact force-displacement. (c) Energy-time.
+As we can see either in Fig. 7 and Tab. 2 a good agreement is obtained between experiments and
+numerical predictions.
+On the one hand, the proposed contact algorithm combined with the proposed
+damage models is able to capture the maximum impact force very well and the maximum displacement
+pretty well with errors below 10%, respectively. On the contrary, the different dissipated energies obtained by
+the experiments show a high dispersity between them, resulting in difficulty in conducting a fair comparison
+between the predicted value 1.1 J and the experimental mean value.
+Experiment
+Prediction
+Difference (%)
+Mean
+Std.
+Maximum impact force, f c
+max (kN)
+5.3
+0.2
+5.2
+-1.0
+Maximum displacement, dmax (mm)
+5.4
+0.1
+4.9
+-9.3
+Dissipated energy, Edis (J)
+0.2
+0.3
+1.1
+>10
+Table 2: Comparison of numerical results with experimental data for the impact case scenario of the plate made of T800S/M21
+material.
+3.2.2. Coupon made of AS4/8552 material
+This second case consists of a coupon made with the AS4/8552 material.
+The plate has a stacking
+sequence of [454/04/ − 454/904]S with a nominal ply thickness of 0.181 mm resulting a plate thickness of
+5.8 mm. This case study has higher energy (19.3 J) than the previous one, and it also includes clusters of four
+and eight plies which are potential for extensive matrix cracks and delaminations. The energy of 19.3 J also
+falls into BVID analysis. The in-plane element sizes used in [17] and [51] are 0.3 mm and 0.5 mm respectively.
+In the present work, two element sizes are studied using the mesh refinement algorithm described in Sec. 2.4,
+see Tab. 3. The base mesh for the plate has a total of 335 622 hexahedron elements (≈ 1 million of Degrees
+Of Freedom (DOF)). The total time for the simulation is set to 5.0 ms. In this case, the striker has a mass
+17
+
+of 5 kg, and its radius is 8 mm. The initial velocity of the striker considering the gap previously mentioned
+is 2.78 m/s.
+Refinement
+Element
+No. element
+No. elem.
+No. nodes
+Initial stable
+level, ndivi
+size (mm)
+through ply cluster
+plate
+plate
+time increment (s)
+0
+0.7250
+1
+335 622
+364 320
+7.378 × 10−8
+1
+0.3625
+2
+2 109 624
+2 219 983
+3.515 × 10−8
+Table 3: Element sizes used on the coupon made of AS4/8552 material system and initial stable time increment for each case
+study.
+The numerical predictions of the impact force-displacement and energy - time curves are shown in Fig.
+8 and Fig. 9 respectively, using different element sizes. The most important physics variables for a proper
+validation are summarized in Tab. 4. This table compares the experimental results from [51] with the
+numerical predictions.
+Case
+f c
+del (kN)
+f c
+max (kN)
+dmax (mm)
+Edis (J)
+Aproj
+del
+(mm2)
+Experiment [17]
+4.41
+7.74
+3.72
+12.03
+3898.3
+Numerical (le = 0.7250 mm)
+4.20
+8.70
+3.60
+7.70
+4723.1
+Numerical (le = 0.3625 mm)
+4.30
+8.30
+3.70
+7.90
+5249.20
+Table 4: Comparison of the numerical results obtained with the proposed framework with experimental data from [51]. fc
+del is
+the delamination threshold force, fc
+max is the maximum contact force, dmax is the maximum indentation, Edis is the dissipated
+energy and Aproj
+del
+is the projected delamination area.
+The initial elastic deflection of the plate is very well captured for all the meshes (Fig. 8), meaning that
+the stiffness of the plate is accurately predicted by the PDN contact algorithm. After that, delamination
+onset occurs at the top of the elastic part, around 4.5 kN. This point is also very well captured by the
+interlaminar damage model using cohesive elements between each of the ply clustering. Then, a combination
+of interlaminar and intralaminar damage occurs until the striker reaches both the maximum load and
+displacement, resulting with a pretty good prediction as also shown in Tab. 4. Since damage appears, the
+continuum damage models and the characterization of the material properties play a fundamental role in the
+simulation of this benchmark case. Despite the delamination threshold, maximum force and displacement
+are very well captured; the dissipated energy and the projected delamination area are overpredicted, see
+Tab. 4. According to Soto et al. [51], the projected delamination and the corresponding energy dissipated
+could be considerably improved when using solid elements with one integration point for the bulk material
+and cohesive contact surfaces instead of cohesive elements to be able to better predict the delamination
+shapes at each interface of the layup.
+18
+
+Figure 8: Numerical prediction of the force-displacement curved using two element sizes and correlation with the experiment
+from Gonz´alez et al. [17].
+Figure 9: Numerical prediction of the impact energy vs. time using two element sizes and correlation with the experiment from
+Gonz´alez et al. [17].
+Fig. 10 depicts and aims to quantify the most important failure mechanisms that appear on the plate.
+Fiber damage is represented by damage variable D1, which includes both fiber breakage and fiber kinking,
+see Fig. 10a. As we can see, this source of damage is not the most predominant and mostly appears at the
+bottom of the striker. Matrix cracking is represented with the damage variable D2, which includes matrix
+tension and compression (Fig. 10b). Finally, the last source of damage is delamination (Fig. 10b). Its
+prediction is compared with the shape obtained from the experiment, which is represented in dashed lines.
+As we discussed previously, this source of damage is overpredicted for all the element sizes studied, see Tab.
+4 and further research would be required in that direction as the values of the material properties, and
+19
+
+the damage models play a fundamental role. Furthermore, the extensive matrix cracks and delamination
+predicted for this impact case scenario corroborate the experimental observations by Gonz´alez et al. [18] on
+the effect of ply clustering to originate extensive matrix cracks and large delaminations.
+3898.3 mm2
+(a)
+(b)
+(c)
+Experiment
+Numerical
+5248.2 mm2
+10 mm
+Dcoh
+D1
+D2
+Figure 10: Numerical prediction of the damage occurred in the coupon. (a) Fiber damage, D1. (b) Matrix cracking, D2. (c)
+Projected delamination, Dcoh. The numerical result correspond to the most refined mesh.
+3.3. Parallel performance
+The speedup and the parallel efficiency of the proposed contact algorithm for solving low-velocity impact
+events are evaluated in this section. All the executions are conducted in MareNostrum4 supercomputer. A
+strong scalability analysis has been conducted using a larger mesh than the ones studied in Sec. 3.2. The
+model corresponds to the AS4/8552 impact case scenario. The new mesh has a total of 74M elements with
+228M of DOF, which results from a base mesh of 1 472 328 elements using two levels of the mesh refinement
+algorithm. Strong scalability consists of fixing the mesh and solving the problem with a different number
+of processors, Central Processing Unit (CPU). The strong speedup is calculated as
+t0
+tN while the parallel
+efficiency is calculated as
+t0N0
+tNN , where N is the number of processors and t0 is the reference simulation
+time for N0 processors. The number of processors used for this analysis ranges from 192 to 2400. Due to
+the resolution of the problem following a multibody/multicode approach, the number of processors for the
+striker is fixed to 16 (sufficiently for its mesh) while the number of processors for the plate is changed. It
+is worth mentioning that the strong computational effort falls in the resolution (deformation) of the plate
+and the localization and exchange of information phases, as explained in [19]. Due to the small time step
+in this simulation, 9.261 × 10−9 s, the simulations for the scalability curve are limited to the first 7460 time
+20
+
+DAM01
+0.0e+00 0.1
+0.2
+0.3
+0.4
+0.5
+0.7
+0.80.9 1.0e+00
+Y
+Y
+L.
+DAM02
+DCOHE
+0.0e+00 0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9 1.0e+00
+0.0e+00 0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9 1.0e+00steps. The end of the execution (last time step) corresponds to an impact force of approximately 1 kN, which
+falls into the linear elastic regime of the force-displacement curve shown in Fig. 8. The strong speedup and
+parallel efficiency are shown in Fig. 11. The ideal scalability and efficiency are represented with a dashed
+line.
+Average No. of elements per core
+Strong speedup
+Parallel Efficiency
+Total No. of processors
+Total No. of processors
+Figure 11: Strong scalability of the low velocity impact test with a plate mesh of 74M hexahedron elements.
+The model
+corresponds to the benchmark case using AS4/8552 material system.
+The results obtained in Fig. 11 show that the scalability of the problem in explicit analysis is really
+good up to 2400 processors using a mesh of 74M elements. The parallel efficiency is maintained above 90%,
+which demonstrates the good scalability of the proposed framework to deal with large-scale problems. This
+linear behavior is also shown in Tab. 5 where we summarize the total CPU time for each execution using a
+different number of processors while maintaing fixed the size of the problem.
+21
+
+No. of CPUs
+192
+384
+768
+1536
+1824
+2064
+2400
+17:20
+08:33
+4:15
+02:11
+01:51
+01:39
+01:27
+Table 5: Total CPU time expressed in hh:mm for different executions of the low-velocity impact simulation considering a
+fixed mesh of 74M of elements (228M of DOF) with a total of 7460 time steps. This CPU time includes the preprocess part,
+where two mesh refinement levels are performed and the solution of the contact problem within the elastic regime of the
+force-displacement curve.
+It is also worth mentioning that the application of the proposed contact algorithm in explicit dynam-
+ics improves both the speedup and the parallel efficiency in comparison to an implicit resolution for the
+deformable body (plate), as already studied by the authors in [19]. This improvement in computational per-
+formance is mainly attributed to the time integration scheme for the deformable body. In explicit dynamics,
+it is not required to invert the global matrix of the system. In this case, the unknown is the acceleration, and
+the system is solved directly using the lumped mass matrix and the global force vector on the right-hand
+side. The reader is referred to [6] for more details.
+4. Conclusions
+In this paper, we apply the parallel PDN contact algorithm to simulate low-velocity impact events on
+fiber-reinforced polymer composites using a High-Performance Computing environment. Existing damage
+models from the literature have been implemented in our multiphysics finite element code Alya to simulate
+the material damage. Moreover, we introduce a new capability in the in-house mesh refinement algorithm to
+deal with cohesive elements and other element types, such as continuum shell elements. This is really attrac-
+tive as we can refine the finite element mesh at the beginning of the simulation with a meager computational
+cost.
+We validate the whole framework with several benchmark tests. The last example corresponds to a
+well-known low-velocity impact test following the ASTM standard for damage resistance analysis. In this
+case, we study two impact case scenarios with two different material systems: the T800S/M21 and the
+AS4/8552, obtaining excellent predictions for impact behavior and pretty good damage occurrence compared
+to experimental data from the literature. Additionally, the mesh refinement algorithm’s capabilities have
+been demonstrated for the plate made of AS4/8552 material.
+Finally, we evaluate the parallel performance of the impact simulation. Despite not using ”very” large
+meshes for the physics validation cases, we have generated a new larger mesh using the mesh refinement
+algorithm. The reason behind this is the stable time increment, which becomes smaller as the element size
+decrease. The new mesh has 74M hexahedron elements (228M of DOF) using full integration. An excellent
+computational efficiency (above 90%) has been obtained up to 2400 CPUs, demonstrating its applicability
+to solve large mesh models ranging from micro-scale to macro-scale.
+22
+
+A further conclusion of this work is that we demonstrate the potential application of the parallel PDN
+contact algorithm for low-velocity impact events and its parallel efficiency for large models compared to
+traditional Penalty or Lagrange contact-based methods. As we commented previously, we use full integration
+elements for all the examples; the use of reduced integration elements, which are more appropriate for
+explicit schemes and overcome the well-known locking pathologies from solid brick elements, can considerably
+increase the speedup of the simulations. Moreover, the localization of contact nodes and the communication
+between subdomains created by the domain decomposition method is a crucial issue for further research as
+it is the main bottleneck regarding the computational efficiency of contact algorithms.
+Acknowledgements
+This work has received funding from the Clean Sky 2 Joint Undertaking (JU) under grant agreements
+No. 807083 and No. 945521 (SHERLOC project). The JU receives support from the European Union’s
+Horizon 2020 research and innovation program and the Clean Sky 2 JU members other than the Union.
+The authors gratefully acknowledge Hellenic Aerospace Industry for manufacturing of the coupons made
+of T800S/M21 material and Kirsa Mu˜noz and Miguel ´Angel Jim´enez from Element Materials Technology
+Seville for conducting the experimental impact tests and providing all the experimental data. A. Quintanas-
+Corominas acknowledges financial support from the European Union-NextGenerationEU and the Ministry
+of Universities and Recovery, Transformation and Resilience Plan of the Spanish Government through a call
+of the University of Girona (grant REQ2021-A-30). G. Guillamet thankfully acknowledges the computer
+resources at MareNostrum and the technical support provided by Barcelona Supercomputing Center (FI-
+2019-2-0010). Last but not least, the authors would also like to thank the late Claudio Lopes for all the
+interesting discussions and contributions to the simulation of impact events and damage on composites.
+Appendix A. Algorithms
+Here we summarize the main algorithms of the whole modeling framework to solve low-velocity impact
+events for damage resistance of fiber-reinforced polymer composites by making use of High-Performance
+Computing.
+23
+
+Algorithm 1 Main code for the partial Dirichlet-Neumann (PDN) contact algorithm.
+This PDN contact algorithm is treated as a coupling problem between two or more body instances.
+In the present algorithm,
+we describe the contact algorithm between two code instances: a rigid body represented by the domain Ωa and the deformable
+body represented by the domain Ωb.
+The coupling is performed through the exchange of boundary conditions at the contact
+interface following a Gauss-Seidel strategy. At each time step, contact detection is done for both instances, and synchronization
+and localization is executed. When contact is detected (at least one boundary node belonging to the deformable body is penetrated
+inside the rigid body), the rigid one computes and sends to the deformable body all the information required for the enforcement
+of the kinematic boundary conditions.
+The reader is referred to the Ph.D. from Rivero [46], or [19] for more details on the
+implementation aspects of the proposed contact algorithm.
+Require: Ωa, Ωb
+1: loop time
+2:
+Compute time step, tn+1
+3:
+loop reset
+4:
+if Rigid body, Ωa then
+5:
+Contact detection (localization)
+▷ Contact detection & localization, Algo. 1 in [19]
+6:
+Exchange data: receive f cont from Ωb
+▷ Exchange & communication data, Algo. 2 in [19]
+7:
+call calculateProjections()
+▷ Projections & local coordinate system, Algo. 3 in [19]
+8:
+call RK4Scheme()
+▷ Solve system
+9:
+Exchange data: send projection data to Ωb
+▷ Exchange & communication data, Algo. 2 in [19]
+10:
+end if
+11:
+if Deformable body, Ωb then
+12:
+Contact detection (localization)
+▷ Contact detection & localization, Algo. 1 in [19]
+13:
+Exchange data: receive data (projections) from Ωa
+▷ Algo. 2 in [19]
+14:
+call EssentialBoundaryCondition()
+▷ Contact nodes & Dirichlet condition, Algo. 4 in [19]
+15:
+call ExplicitScheme()
+▷ Solve system
+16:
+call ReleaseNodes()
+▷ Algo. 2
+17:
+Exchange: send f cont to Ωa
+▷ Exchange & communication data, Algo. 2 in [19]
+18:
+end if
+19:
+if kfl reset = 0 then
+20:
+exit loop reset
+21:
+end if
+22:
+end loop
+23: end loop
+Algorithm 2 ReleaseNodes() algorithm for explicit time integration schemes
+This algorithm is executed concurrently and for each subdomain at the end of the time step tn+1. The kfl reset is the key flag for
+the repetition of the current time step tn+1 when exists adhesion contact nodes. The sign of the contact force is checked according
+to Eq. 2. The key flag to release the adhesion nodes is called kfl nodes to release. Then the adhesion contact nodes are released
+(as free non-contacting nodes), and the time step is repeated, activating the reset key flag. As all the subdomains require to know
+if the time step has to be repeated or not, the MPI_MAX is in charge to collect the value of the reset for all the subdomains of the
+mesh.
+1: kfl reset ← 0
+2: Get contact force f c and mark adhesion nodes
+3: if kfl nodes to release then
+4:
+Adhesion nodes are set to free nodes
+5:
+kfl reset ← 1
+6: end if
+7: call MPI_MAX(kfl reset)
+24
+
+Algorithm 3 Recursive mesh multiplication algorithm
+The level of mesh refinement is set with the parameter ndivi. ne, nn and nb are the total number of elements, nodes and boundaries
+of the new mesh. The same parameters with the superscript 0 indicate the initial dimension of the mesh. In order to define the
+dimensions of the new mesh is necessary to know the total number of edges (nedgg) and faces nfacg of the initial mesh. Then,
+once the dimensions are known, the DivideMesh() subroutine is in charge of doing the following actions: i) divide each edge and
+face from the initial mesh, ii) define the new element connectivities, iv) define the new element boundary connectivities and v)
+assign the material codes and the corresponding fields such as material coordinate systems. The last step of the mesh division
+algorithm is to reconstruct the interface domains through the ReconstructInterfaceDomains() subroutine. The reader is referred
+to Houzeuax et al. [22] for more details on the implementation and parallel aspects of the proposed algorithm.
+Input n0
+e, n0
+n, n0
+b , ndivi
+Output ne, nn, nb
+1: for idivi = 1, ndivi do
+2:
+nedgg = GetEdges()
+3:
+nfacg = GetFaces()
+4:
+ne, nn, nb = GetDimensions()
+▷ Eq. 14
+5:
+call DivideMesh()
+▷ Sec. 3.2 [22]
+6:
+call ReconstructInterfaceDomains()
+▷ Sec. 3.2 [22]
+7: end for
+Algorithm 4 Workflow of the intralaminar damage model.
+The strain and stress tensors, ε and σ, are defined in the material coordinate system using compact notation [6]. The superscripts
+n and n + 1 define the past and current time steps, respectively. The subscripts N indicate the four damage mechanisms associated
+with the loading function φN and internal threshold variables rN (fibre breaking, fibre kinking, tensile matrix cracking, and
+compressive matrix cracking).
+In turn, the subscript M indicates the five uniaxial damage states DM, represented in 2.
+The
+required material properties are i) elastic properties (E11, E22, ν12, ν23, G12 ), ii) ply strengths (XT , XC, YT , YC, SL), iii) fracture
+toughness (GXT , GXC, GY T , GY C, GSL) associated with the damage mechanism, and iv) yield strength and hardening (Sp, Kp);
+all these properties can be obtained through standardised tests or computational micromechanics simulations [30]. The required
+parameters are: characteristic element length ℓc [33] and state variables at the past time step, i.e. εn
+p , and rt
+M. At the initial time
+step, the state variables are initialised as εn
+p = 0 and rn
+M = 1.
+Input εn+1, εn
+p , rn
+M, ℓc, material properties
+Output σn+1, εn+1
+p
+, rn+1
+M
+1: εn+1
+p
+(εn
+p )
+▷ Plastic strains, yield function in [51]
+2: εn+1
+e
+← εn+1 − εn+1
+p
+▷ Effective elastic strains
+3: σn+1
+e
+← H−1 · εn+1
+e
+▷ Effective compliance matrix H in [30]g
+4: φn+1
+M
+(σn+1
+e
+)
+▷ Loading functions (failure criteria), Eqs. 8, 13, 20, 21 in [32]
+5: rn+1
+N
+(φn+1
+N
+, rn
+N)
+▷ Damage thresholds, Eqs. 24, 26 in [32]
+6: Dn+1
+M
+(rn+1
+N
+)
+▷ Damage state variables according [51] and Eq 6 in [32]
+7: σn+1 ← H−1(Dn+1
+M
+) · εn+1
+e
+▷ Nominal compliance matrix H(Dn+1
+M
+) in [30]
+25
+
+Algorithm 5 Workflow of the cohesive zone model.
+The displacement jumps and interface tractions, ∆ = {∆1, ∆2, ∆3}T and τ = {τ1, τ2, τ3}T , are defined at the mid-plane being
+1 and 2 tangential and 3 normal directions. The superscripts t and t + 1 define the past and current time steps, respectively. In
+turn, the subscript M indicates the pure-mode I and II openings associated with the opening directions, I ↔ {3} and II ↔ {1, 2}.
+The latter is also referred with the subscript sh in [56]. The required input parameters are i) onset displacement jumps (∆Mo),
+ii) critical displacement jumps (∆Mc), iii) penalty stiffness (KM), and iv) Benzeggagh-Kenane exponent for the mixed-mode ratio
+(η). The onset and critical jumps can be obtained from the cohesive strengths (τM) and fracture toughness material properties by
+∆Mo = τM/KM and ∆Mc = 2GM/τM, respectively. The damage threshold state variable at the past time rn
+D, which is initialised
+at the initial time step as rn
+D = 0, is also required to evaluate the model.
+Input ∆n+1, rn
+D, material properties
+Output τ n+1, rn+1
+D
+1: Kn+1
+B
+(∆n+1)
+▷ Local mixed-mode penatly stiffness, Eq. 13 in [56]
+2: Bn+1(∆n+1)
+▷ Local mixed-mode ratio, Eq. 17 in [56]
+3: λn+1
+o
+(Bn+1, Kn+1
+B
+)
+▷ Local mixed-mode onset jump, Eq. 26 in [56]
+4: λn+1
+c
+(Bn+1, Kn+1
+B
+, λn+1
+o
+)
+▷ Local mixed-mode propagation jump, Eq. 24 in [56]
+5: λt+1(∆t+1)
+▷ Local mixed-mode equivalent jump, Eq. 12 in [56]
+6: Hn+1(λn+1, λn+1
+o
+, λn+1
+c
+)
+▷ Loading function (failure criteria), Eq. 20 in [56]
+7: rn+1
+D
+(Hn+1, rn
+D)
+▷ Damage threshold, Eq. 21 in [56]
+8: Dn+1(rn+1
+D
+, λn+1
+o
+, λn+1
+c
+)
+▷ Damage state, Eq. 20 in [56]
+9: τ n+1
+coh (Dn+1, ∆t+1)
+▷ Cohesive tractions, Eq. 7 in [56]
+10: τ n+1
+con (∆n+1)
+▷ Contact tractions, Eq. 8 in [56]
+11: τ n+1 ← τ n+1
+coh + τ n+1
+con
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+

FNE0T4oBgHgl3EQfQwDu/content/tmp_files/2301.02199v1.pdf.txt

@@ -0,0 +1,684 @@
+arXiv:2301.02199v1  [math.GR]  5 Jan 2023
+On the Generalized Fitting Height and Nonsoluble
+Length of the Mutually Permutable Products of Finite
+Groups∗
+Viachaslau I. Murashka1,2 and Alexander F. Vasil’ev3,2
+Abstract
+The generalized Fitting height h∗(G) of a finite group G is the least number h such
+that F∗
+h(G) = G, where F∗
+(0)(G) = 1, and F∗
+(i+1)(G) is the inverse image of the generalized
+Fitting subgroup F∗(G/F∗
+(i)(G)). Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G
+be the shortest normal series in which for i odd the factor Gi+1/Gi is p-soluble (possibly
+trivial), and for i even the factor Gi+1/Gi is a (non-empty) direct product of nonabelian
+simple groups. Then h = λp(G) is called the non-p-soluble length of a group G. We proved
+that if a finite group G is a mutually permutable product of of subgroups A and B then
+max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1 and max{λp(A), λp(B)} = λp(G).
+Also we introduced and studied the non-Frattini length.
+Keywords: Finite group; generalized Fitting subgroup; mutually permutable product
+of groups; generalized Fitting height; non-p-soluble length; Plotkin radical.
+1
+Introduction and the Main Results
+All groups considered here are finite. E.I. Khukhro and P. Shumyatsky introduced and
+studied interesting invariants of a group: the generalized Fitting height and the nonsoluble
+length [11–13]. The first one is the extension of the well known Fitting height to the class of
+all groups and the second one implicitly appeared in [8,20].
+Definition 1.1 (Khukhro, Shumyatsky). (1) The generalized Fitting height h∗(G) of a finite
+group G is the least number h such that F∗
+h(G) = G, where F∗
+(0)(G) = 1, and F∗
+(i+1)(G) is the
+inverse image of the generalized Fitting subgroup F∗(G/F∗
+(i)(G)).
+(2) Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series
+in which for i odd the factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor
+Gi+1/Gi is a (non-empty) direct product of nonabelian simple groups. Then h = λp(G) is called
+the non-p-soluble length of a group G.
+(3) Recall that λ2(G) = λ(G) is the nonsoluble length of a group G.
+In [12] E.I. Khukhro and P. Shumyatsky showed that in the general case the generalized
+Fitting height of a factorized group is not bounded in terms of the generalized Fitting heights
+of factors. The same situation is also for the nonsoluble length.
+Recall [1, Definition 4.1.1] that a group G is called a mutually permutable product of its
+subgroups A and B if G = AB, A permutes with every subgroup of B and B permutes with
+1email: [email protected]
+2Francisk Skorina Gomel State University, Gomel, Belarus
+3email: [email protected]
+∗Supported by BFFR Φ23PHΦ-237
+1
+
+every subgroup of A. The products of mutually permutable subgroups is the very interesting
+topic of the theory of groups (for example, see [1, Chapter 4]).
+The main result of our paper is
+Theorem 1.1. Let a group G be the product of the mutually permutable subgroups A and B.
+Then
+(1) max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1.
+(2) max{λp(A), λp(B)} = λp(G) for any prime p. In particular, max{λ(A), λ(B)} = λ(G).
+If a group G is soluble, then h∗(G) = h(G) is the Fitting height of a group G.
+Corollary 1.2 ([10]). If a soluble group G is the product of the mutually permutable subgroups
+A and B, then max{h(A), h(B)} ≤ h(G) ≤ max{h(A), h(B)} + 1.
+Example 1.1. Note that the symmetric group S3 of degree 3 is the mutually permutable product
+of the cyclic groups Z2 and Z3 of orders 2 and 3 respectively. Hence h∗(S3) = max{h∗(Z2), h∗(Z3)}+
+1 = max{h(Z2), h(Z3)} + 1.
+2
+The Functorial Method
+According to B.I. Plotkin [15] a functorial is a function γ which assigns to each group G its
+characteristic subgroup γ(G) satisfying f(γ(G)) = γ(f(G)) for any isomorphism f : G → G∗.
+We are interested in functorials with some properties:
+(F1) f(γ(G)) ⊆ γ(f(G)) for every epimorphism f : G → G∗.
+(F2) γ(N) ⊆ γ(G) for every N ⊴ G.
+(F3) γ(G) ∩ N ⊆ γ(N) for every N ⊴ G.
+Remark 2.1. (0) Functions F∗ and Rp that assign to every group respectively its the generalized
+Fitting subgroup and the p-soluble radical are examples of functorials. It is well known that they
+satisfy (F1), (F2), (F3).
+(1) Recall that a functorial γ is called a Plotkin radical if it satisfies (F1), idempotent (i.e.
+γ(γ(G)) = γ(G)) and N ⊆ γ(G) for every γ(N) = N ⊴ G [5, p. 28].
+(2) A functorial that satisfies (F3) is often called hereditary (nevertheless, the same word
+means different in the theory of classes of groups).
+(3) A functorial γ is a hereditary Plotkin radical if and only if it satisfies (F1), (F2), (F3).
+Let prove it. Assume that γ is a hereditary Plotkin radical. We need only to prove that it satisfies
+(F2). If N ⊴ G, then γ(N) char N ⊴ G. So γ(N) ⊴ G. Now γ(N) = γ(γ(N)) ⊆ γ(G). Thus
+a hereditary Plotkin radical satisfies (F1), (F2), (F3). Assume that γ satisfies (F1), (F2), (F3).
+We need only to prove that it is idempotent. By (F3) we have γ(G) = γ(G) ∩ G ⊆ γ(γ(G)) ⊆
+γ(G). Thus γ(γ(G)) = γ(G).
+(4) The functorial Φ which assigns to every group G its Frattini subgroup Φ(G) satisfies
+(F1) and (F2) but not (F3).
+(5) If γ satisfies (F2) and (F3), then γ(G) ∩ N = γ(N) for every group G and N ⊴ G.
+Lemma 2.1. If γ satisfies (F1) and (F2), then γ(G1 × G2) = γ(G1) × γ(G2) for any groups
+G1 and G2.
+Proof. From Gi ⊴ G1 × G2 it follows that γ(Gi) ⊆ γ(G1 × G2) by (F2) for i ∈ {1, 2}. Note
+that γ(G1 × G2)Gi/Gi ⊆ γ((G1 × G2)/Gi) = (γ(G¯i) × Gi)/Gi by (F1) for i ∈ {1, 2}. Now
+γ(G1 × G2) ⊆ (γ(G1 × G2)G2) ∩ (γ(G1 × G2)G1) ⊆
+(γ(G1) × G2) ∩ (G1 × γ(G2)) = γ(G1) × γ(G2).
+Thus γ(G1 × G2) = γ(G1) × γ(G2).
+2
+
+Recall [15] that for functorials γ1 and γ2 the upper product γ2 ⋆ γ1 is defined by
+(γ2 ⋆ γ1)(G)/γ2(G) = γ1(G/γ2(G)).
+Proposition 2.2. Let γ1 and γ2 be functorials. If γ1 and γ2 satisfy (F1) and (F2), then γ2 ⋆γ1
+satisfies (F1) and (F2). Moreover if γ1 and γ2 also satisfy (F3), then γ2 ⋆ γ1 satisfies (F3).
+Proof. (1) γ2 ⋆ γ1 satisfies (F1).
+Let f : G → f(G) be an epimorphism. From f(γ2(G)) ⊆ γ2(f(G)) it follows that the
+following diagram is commutative.
+G
+f
+�
+f4
+�P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+f1
+�
+f(G)
+f3
+�
+G/γ2(G)
+f2� f(G)/γ2(f(G))
+Let X = γ1(G/γ2(G)) and Y = γ1(f(G)/γ2(f(G))). Note that (γ2 ⋆ γ1)(G) = f −1
+1 (X) and
+(γ2 ⋆ γ1)(f(G)) = f −1
+3 (Y ) by the definition of γ2 ⋆ γ1. Since γ1 satisfies (F1), we see that
+f2(X) ⊆ Y . Hence X ⊆ f −1
+2 (Y ). Now (γ2 ⋆ γ1)(G) ⊆ f −1
+1 (f −1
+2 (Y )) = f −1
+4 (Y ). So
+f((γ2 ⋆ γ1)(G)) ⊆ f(f −1
+4 (Y )) = f −1
+3 (Y ) = (γ2 ⋆ γ1)(f(G)).
+Thus γ2 ⋆ γ1 satisfies (F1).
+(2) γ2 ⋆ γ1 satisfies (F2).
+Let N ⊴ G. From γ2(N) char N ⊴ G it follows that γ2(N) ⊴ G. Since γ2 satisfies (F2),
+we see that γ2(N) ⊆ γ2(G). So the following diagram is commutative.
+G
+f1�
+f3
+�❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+G/γ2(N)
+f2
+�
+G/γ2(G)
+Let X = γ1(G/γ2(N)), Y = γ1(N/γ2(N)) and Z = γ1(G/γ2(G)). Note that (γ2 ⋆ γ1)(G) =
+f −1
+3 (Z) and (γ1 ⋆ γ2)(N) ⊆ f −1
+1 (Y ). Since γ1 satisfies (F1) and (F2), we see that f2(X) ⊆ Z
+and Y ⊆ X. Now
+(γ2 ⋆ γ1)(N) ⊆ f −1
+1 (Y ) ⊆ f −1
+1 (X) ⊆ f −1
+1 (f −1
+2 (Z)) = f −1
+3 (Z) = (γ2 ⋆ γ1)(G).
+Hence γ2 ⋆ γ1 satisfies (F2).
+(3) If γ1 and γ2 also satisfy (F3), then γ2 ⋆ γ1 satisfies (F3).
+Assume that γ1 and γ2 satisfy (F2) and (F3). Let N ⊴ G.
+Since Nγ2(G)/γ2(G) ∩ (γ2 ⋆ γ1)(G)/γ2(G) ⊴ (γ2 ⋆ γ1)(G)/γ2(G) = γ1(G/γ2(G)), we see by
+(5) of Remark 2.1 that
+γ1((Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G)) = (Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G).
+Note that
+(Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G) =
+(N ∩ (γ2 ⋆ γ1)(G))γ2(G)/γ2(G) ≃ (N ∩ (γ2 ⋆ γ1)(G))/(N ∩ γ2(G))
+= (N ∩ (γ2 ⋆ γ1)(G))/γ2(N) ⊴ N/γ2(N).
+It means that (N ∩ (γ2 ⋆ γ1)(G))/γ2(N) ⊆ γ1(N/γ2(N)). Thus N ∩ (γ2 ⋆ γ1)(G) ⊆ (γ2 ⋆ γ1)(N),
+i.e γ2 ⋆ γ1 satisfies (F3).
+3
+
+Here we introduce the height hγ(G) of a group G which corresponds to a given functorial γ.
+Definition 2.1. Let γ be a functorial. Then the γ-series of G is defined starting from γ(0)(G) =
+1, and then by induction γ(i+1)(G) = (γ(i) ⋆ γ)(G) is the inverse image of γ(G/γ(i)(G)). The
+least number h such that γ(h)(G) = G is defined to be γ-height hγ(G) of G. If there is no such
+number, then hγ(G) = ∞.
+The following Lemma directly follows from Proposition 2.2.
+Lemma 2.3. Let γ be a functorial. If γ satisfies (F1) and (F2), then γ(n) satisfies (F1) and
+(F2) for all natural n. Moreover if γ satisfies (F3), then γ(n) satisfies (F3) for all natural n.
+Lemma 2.4. Let γ be a functorial. If γ satisfies (F1) and (F2), then hγ(G/N) ≤ hγ(G) ≤
+hγ(N) + hγ(G/N) for every N ⊴ G. Moreover, if γ also satisfies (F3), then hγ(N) ≤ hγ(G).
+Proof. Note that γ(n) satisfies (F1) and (F2) for every n by Lemma 2.3.
+Since γ(n) satisfies (F1), G/N = γhγ(G)(G)/N ≤ γ(hγ(G))(G/N) ≤ G/N. So γ(hγ(G))(G/N) =
+G/N. Thus hγ(G/N) ≤ hγ(G).
+Since γ(n) satisfies (F2), we see that N = γ(hγ(N))(N) ⊆ γ(hγ(N))(G). Note that hγ(G/γ(hγ(N))(G)) ≤
+hγ(G/N). Thus hγ(G) ≤ hγ(N) + hγ(G/N).
+Assume that γ also satisfies (F3).
+Then γ(n) satisfies (F3) by Lemma 2.3.
+Now N =
+G ∩ N = γ(hγ(G))(G) ∩ N ⊆ γ(hγ(G))(N) ≤ N. So γ(hγ(G))(N) = N. Thus hγ(N) ≤ hγ(G).
+If γ = F∗, then hγ(G) = h∗(G) for every group G. The non-p-soluble length can also be
+defined with the help of functorials. Here by Rp(G) we denote the p-soluble radical of a group
+G.
+Lemma 2.5. Let Fp = Rp ⋆F∗ ⋆Rp and G be a non-p-soluble group. Then λp(G) is the smallest
+natural i with Fp(i)(G) = G.
+Proof. Let 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series in which for i odd the
+factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor Gi+1/Gi is a (non-empty)
+direct product of nonabelian simple groups.
+Note that G1 ≤ Rp(G) and G2/G1 is quasinilpotent. Hence G2Rp(G)/Rp(G) is quasinilpo-
+tent. It means that G2Rp(G)/Rp(G) ≤ F∗(G/Rp(G)). Hence G2 ≤ (Rp ⋆ F∗)(G). Since G3/G2
+is p-soluble, we see that G3(Rp ⋆ F∗)(G)/(Rp ⋆ F∗)(G) is p-soluble. Hence G3(Rp ⋆ F∗)(G)/(Rp ⋆
+F∗)(G) ≤ Rp(G/(Rp ⋆ F∗)(G)). It means that G3 ≤ Fp(G) = Fp(1)(G).
+Assume that we proved G2i+1 ≤ Fp(i)(G). Let prove that G2(i+1)+1 ≤ Fp(i+1)(G).
+From G2i+1 ≤ Fp(i)(G) it follows that G2i+1 ≤ (Fp(i) ⋆ Rp)(G).
+Note that G2i+2/G2i+1
+is quasinilpotent. It means that G2i+2(Fp(i) ⋆ Rp)(G)/(Fp(i) ⋆ Rp)(G) is quasinilpotent. Hence
+G2i+2 ≤ ((Fp(i)⋆Rp)⋆F∗)(G). Since G2(i+1)+1/G2i+2 is p-soluble, we see that G2(i+1)+1(Fp(i)⋆Rp⋆
+F∗)(G)/(Fp(i)⋆Rp⋆F∗)(G) is p-soluble. Hence G2(i+1)+1(Fp(i)⋆Rp⋆F∗)(G)/((Fp(i)⋆Rp⋆F∗)(G) ≤
+Rp(G/(Fp(i) ⋆ Rp ⋆ F∗)(G)). It means that G2(i+1)+1 ≤ (Fp(i) ⋆ Rp ⋆ F∗ ⋆ Rp)(G) = Fp(i+1)(G).
+Therefore λp(G) ≥ n where n is the smallest integer with Fp(n)(G) = n. Since Rp ⋆Rp = Rp,
+we see that Fp(n)(G) presents a normal series 1 ≤ F1 ≤ F2 ≤ · · · ≤ F2n+1 in which for i odd the
+factor Fi+1/Fi is p-soluble (possibly trivial), and for i even the factor Fi+1/Fi is a (non-empty)
+direct product of nonabelian simple groups. So λp(G) ≤ n. Thus λp(G) = n.
+Now we are able to estimate the γ-height of the direct product subgroups and of the join
+of subnormal subgroups:
+4
+
+Theorem 2.6. Let γ be a functorial with γ(H) > 1 for every group H that satisfies (F1)
+and (F2).
+(1) If G = ×n
+i=1Ai is the direct product of its normal subgroups Ai, then hγ(G) = max{hγ(Ai) |
+1 ≤ i ≤ n}.
+(2) Let G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai. Then hγ(G) ≤
+max{hγ(Ai) | 1 ≤ i ≤ n}. If γ satisfies (F3), then hγ(G) = max{hγ(Ai) | 1 ≤ i ≤ n}.
+Proof. Note that γ(n) satisfies (F1) and (F2) for every n by Proposition 2.2.
+(1) From Lemma 2.1 it follows that if G = ×n
+i=1Ai, then γ(n)(G) = ×n
+i=1γ(n)(Ai). It means
+that hγ(G) = max{hγ(Ai) | 1 ≤ i ≤ n}.
+(2) Assume that G = ⟨Ai | 1 ≤ i ≤ n⟩ is the join of its subnormal subgroups Ai, h1 =
+max{hγ(Ai) | 1 ≤ i ≤ n} and h2 = hγ(G). Since γ(n) satisfies (F2), we see that γ(n)(N) ⊆
+γ(n)(G) for every subnormal subgroup N of G and every n. Now
+G = ⟨Ai | 1 ≤ i ≤ n⟩ = ⟨γ(h1)(Ai) | 1 ≤ i ≤ n⟩ ⊆ γ(h1)(G) ⊆ G.
+Hence γ(h1)(G) = G. It means that h2 ≤ h1.
+Suppose that γ satisfies (F3). Now γ(n) satisfies (F3) for every n by Proposition 2.2. From
+(5) of Remark 2.1 it follows that γ(n)(G) ∩ N = γ(n)(N) for every subnormal subgroup N of G.
+Now Ai = Ai ∩ G = Ai ∩ γ(h2)(G) = γ(h2)(Ai). It means that hγ(Ai) ≤ h2 for every i. Hence
+h1 ≤ h2. Thus h1 = h2.
+Corollary 2.7. Let a group G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.
+Then h∗(G) = max{h∗(Ai) | 1 ≤ i ≤ n} and λp(G) = max{λp(Ai) | 1 ≤ i ≤ n}.
+3
+The Classes of Groups Method
+Recall that a formation is a class F of groups with the following properties: (a) every
+homomorphic image of an F-group is an F-group, and (b) if G/M and G/N are F-groups, then
+also G/(M ∩ N) ∈ F. Recall that the F-residual of a group G is the smallest normal subgroup
+GF of G with G/GF ∈ F.
+A formation is called Fitting if (a) from N ⊴ G ∈ F it follows that N ∈ F and (b) a group
+G ∈ F whenever it is a product of normal F-subgroups. Recall that the F-radical GF of a group
+G is the greatest normal F-subgroup.
+The classes N∗ of all quasinilpotent groups and Sp of all p-soluble groups are Fitting for-
+mations.
+From [3, IX, Remarks 1.11 and Theorem 1.12] and [3, IV, Theorem 1.8] follows
+Lemma 3.1. Let F and H be non-empty Fitting formations. Then
+FH = (G | GF ∈ H) = (G | G/GH ∈ F)
+is a Fitting formation.
+Corollary 3.2. The class Hp = (G | Fp(G) = G) is a Fitting formation.
+It is straightforward to check that for a Fitting formation F, the F-radical can be considered
+as a functorial γ which satisfies (F1), (F2) and (F3). For convenience in this case denote hγ by
+hF. Now h∗(G) = hF∗(G) = hN∗(G) and for a non-p-soluble group λp(G) = hFp(G) = hHp(G).
+Lemma 3.3. Let F be a Fitting formation. If H ̸= 1 and hF(H) < ∞, then hF(HF) = hF(H)−1.
+Proof. Let prove that if H ̸= 1 and hF(H) < ∞, then hF(HF) = hF(H) − 1. Let hF(H) = n
+and hF(HF) = k. Then HF(n−1)(H) < H and H/HF(n−1) ∈ F. It means that HF ≤ HF(n−1).
+Since HF(n−1) satisfies (F3), we see that (HF)F(n−1) = HF. Hence k ≤ n − 1.
+Note that HF = (HF)F(k) ≤ HF(k). It means that H/HF(k) ∈ F. Hence k ≥ n − 1. Thus
+k = n − 1.
+5
+
+If F, H, K ̸= ∅ are formations, then (FH)K = F(HK) by [3, IV, Theorem 1.8]. That is why
+the class Fn = F . . . F
+� �� �
+n
+is a well defined formation.
+Lemma 3.4. For a natural number n and a Fitting formation F holds Fn = (G | hF(G) ≤ n).
+Proof. From Lemma 3.3 it follows that if G ∈ (G | hF(G) ≤ n), then GFn = 1. It means that
+(G | hF(G) ≤ n) ⊆ Fn. Assume that there is a group G ∈ Fn with hF(G) > n. Note that
+G
+F ̸= G for every quotient group G ̸≃ 1 of G. Then hF(GFn) > 0 by Lemma 3.3. It means that
+GFn ̸= 1, a contradiction. Therefore Fn ⊆ (G | hF(G) ≤ n). Thus Fn = (G | h(G) ≤ n).
+In the next lemma we recall the key properties of mutually permutable products
+Lemma 3.5. Let a group G = AB be a mutually permutable product of subgroups A and B.
+Then
+(1) [1, Lemma 4.1.10] G/N = (AN/N)(BN/N) is a mutually permutable product of sub-
+groups AN/N and BN/N for every normal subgroup N of G.
+(2) [1, Lemma 4.3.3(4)] If N is a minimal normal subgroup of a group G, then {N ∩A, N ∩
+B} ⊆ {1, N}.
+(3) [1, Lemma 4.3.3(5)] If N is a minimal normal subgroup of G contained in A and B∩N =
+1, then N ≤ CG(A) or N ≤ CG(B). If furthermore N is not cyclic, then N ≤ CG(B).
+(4) [1, Theorem 4.3.11] AGBG ̸= 1.
+(5) [1, Corollary 4.1.26] A′ and B′ are subnormal in G.
+Recall that π(G) is the set of all prime divisors of |G|, π(F) = ∪
+G∈Fπ(G) and Nπ denote the
+class of all nilpotent π-groups.
+Lemma 3.6. Let F be a Fitting formation. Assume that hF(G) ≤ h + 1 for every mutually
+permutable product G of two F-subgroups. Then
+max{hF(A), hF(B)} − 1 ≤ hF(G) ≤ max{hF(A), hF(B)} + h
+for every mutually permutable product G of two subgroups A and B with hF(A), hF(B) < ∞.
+Proof. If A = 1 or B = 1, then there is nothing to prove. Assume that A, B ̸= 1. Let a group
+G = AB be the product of mutually permutable subgroups A and B. From hF(A), hF(B) < ∞
+it follows that π(G) ⊆ π(F). According to [3, IX, Lemma 1.8] Nπ(F) ⊆ F. Note that A′ and B′
+are subnormal in G by (5) of Lemma 3.5. Since HF ⊴ HNπ(F) ⊴ H′ holds for every π(F)-group
+H, subgroups AF and BF are subnormal in G. Let C = ⟨AF, BF⟩G = ⟨{(AF)x | x ∈ G}∪{(BF)x |
+x ∈ G}⟩. Then by (2) of Theorem 2.6 and by Lemma 3.3
+hF(C) = max
+�
+{(hF(AF)x) | x ∈ G} ∪ {(hF(BF)x) | x ∈ G}
+�
+= max{hF(AF), hF(BF)} = max{hF(A), hF(B)} − 1.
+Now G/C = (AC/C)(BC/C) is a mutually permutable product of F-subgroups AC/C and
+BC/C by (1) of Lemma 3.5. It means that hF(G/C) ≤ h + 1 by our assumption. With the
+help of Lemma 2.4 we see that
+hF(G) ≤ hF(C) + hF(G/C) ≤ max{hF(A), hF(B)} − 1 + 1 + h = max{hF(A), hF(B)} + h.
+From the other hand, hF(G) ≥ hF(C) = max{hF(A), hF(B)} − 1 by (2) of Theorem 2.6.
+Lemma 3.7. Let F be a Fitting formation. Assume that a group G is the least order group
+with
+(1) G is a mutually permutable product of two subgroups A and B with hF(A) ≥ hF(B);
+(2) hF(G) = hF(A) − 1.
+Then G has the unique minimal normal subgroup N, N ≤ A and hF(A/N) = hF(A) − 1.
+6
+
+Proof. Let N be a minimal normal subgroup of G. Then N ∩ A ∈ {N, 1} by (2) of Lemma 3.5.
+Assume that N ∩ A = 1. Now G/N = (AN/N)(BN/N) is a mutually permutable product
+of groups AN/N and BN/N by (1) of Lemma 3.5. By our assumption and hF(G) ≥ hF(G/N) ≥
+hF(AN/N) = hF(A), a contradiction. Hence N ∩ A = N for every minimal normal subgroup N
+of G.
+Now hF(G) + 1 = hF(A) > hF(G) ≥ hF(G/N) ≥ hF(A/N) ≥ hF(A) − 1. It means that
+hF(G) = hF(A/N) = hF(A) − 1.
+If G has two minimal normal subgroups N1 and N2, then hF(A/N1) = hF(A/N2) = hF(A)−1.
+It means hF(A) < hF(A) − 1 by Lemma 3.4, a contradiction. Hence G has a unique minimal
+normal subgroup N.
+4
+Proof of Theorem 1.1(1)
+Our proof relies on the notion of the X-hypercenter. A chief factor H/K of G is called
+X-central in G provided
+(H/K) ⋊ (G/CG(H/K)) ∈ X
+(see [18, p. 127–128] or [7, 1, Definition 2.2]). A normal subgroup N of G is said to be X-
+hypercentral in G if N = 1 or N ̸= 1 and every chief factor of G below N is X-central. The
+symbol ZX(G) denotes the X-hypercenter of G, that is, the product of all normal X-hypercentral
+in G subgroups. According to [18, Lemma 14.1] or [7, 1, Theorem 2.6] ZX(G) is the largest
+normal X-hypercentral subgroup of G. If X = N is the class of all nilpotent groups, then
+ZN(G) = Z∞(G) is the hypercenter of G.
+Lemma 4.1. Let n be a natural number.
+Then (N∗)n = (G | h∗(G) ≤ n) = (G | G =
+Z(N∗)n(G)).
+Proof. First part follows from Lemma 3.4. It is well known that the class of all quasinilpotent
+groups is a composition (or Baer-local, or solubly saturated) formation (see [2, Example 2.2.17]).
+According to [18, Theorem 7.9] (N∗)n is a composition formation. Now (N∗)n = (G | G =
+Z(N∗)n(G)) by [7, 1, Theorem 2.6].
+For a normal section H/K of G the subgroup C∗
+G(H/K) = HCG(H/K) is called an inneriser
+(see [2, Definition 1.2.2]). It is the set of all elements of G that induce inner automorphisms
+on H/K.
+From the definition of the generalized Fitting subgroup it follows that it is the
+intersection of innerisers of all chief factors.
+Lemma 4.2. Let N be a normal subgroup of a group G. If N is a direct product of isomorphic
+simple groups and h∗(G/C∗
+G(N)) ≤ k − 1, then F∗
+(k)(G/N) = F∗
+(k)(G)/N.
+Proof. Assume that h∗(G/C∗
+G(N)) ≤ k − 1.
+Let F/N = F∗
+(k)(G/N).
+Then F∗
+(k)(G) ⊆ F.
+Now F/C∗
+F(N) ≃ FC∗
+G(N)/C∗
+G(N) ⊴ G/C∗
+G(N). Therefore h∗(F/C∗
+F(N)) ≤ k − 1. It means
+that h∗(F/C∗
+F(H/K)) ≤ k − 1 for every chief factor H/K of F below N. Hence (H/K) ⋊
+(F/CF(H/K)) ∈ (N∗)k for every chief factor H/K of F below N. It means that N ≤ Z(N∗)k(F).
+Thus F ∈ (N∗)k by Lemma 4.1. So F ⊆ F∗
+(k)(G). Thus F∗
+(k)(G) = F.
+Lemma 4.3. If a group G = AB is a product of mutually permutable quasinilpotent subgroups
+A and B, then h∗(G) ≤ 2.
+Proof. To prove this lemma we need only to prove that if a group G = AB is a product
+of mutually permutable quasinilpotent subgroups A and B, then G ∈ (N∗)2 by Lemma 4.1.
+Assume the contrary. Let G be a minimal order counterexample.
+(1) G has a unique minimal normal subgroup N and G/N ∈ (N∗)2.
+7
+
+Note that G/N is a mutually permutable product of quasinilpotent subgroups (AN/N) and
+(BN/N) by (1) of Lemma 3.5. Hence G/N ∈ (N∗)2 by our assumption. Since (N∗)2 is a
+formation, we see that G has a unique minimal normal subgroup. According to (4) of Lemma
+3.5 AGBG ̸= 1. WLOG we may assume that G has a minimal normal subgroup N ≤ A.
+(2) N ≤ A ∩ B.
+Suppose that N ∩ B = 1. Then A ≤ CG(N) or B ≤ CG(N) by (3) of Lemma 3.5. If A ≤
+CG(N), then N ⋊ G/CG(N) ≃ N ⋊ B/CB(N) ∈ (N∗)2. If B ≤ CG(N), then N ⋊ G/CG(N) ≃
+N ⋊ A/CA(N) ∈ (N∗) ⊆ (N∗)2 by [2, Corollary 2.2.5]. In both cases N ≤ Z(N∗)2(G). It means
+that G ∈ (N∗)2, a contradiction. Now N ∩ B ̸= 1. Hence N ≤ A ∩ B by (2) of Lemma 3.5.
+(3) N is non-abelian.
+Assume that N is abelian. Since A is quasinilpotent, we see that A/CA(N) is a p-group.
+By analogy B/CB(N) is a p-group. Note that A/CA(N) ≃ ACG(N)/CG(N) and B/CB(N) ≃
+BCG(N)/CG(N). From G = AB it follows that G/CG(N) is a p-group. Since N is a chief
+factor of G, we see that G/CG(N) ≃ 1. So N ≤ Z∞(G) ≤ Z(N∗)2(G). Thus G ∈ (N∗)2, a
+contradiction. It means that N is non-abelian.
+(4) The final contradiction.
+Now N is a direct product of minimal normal subgroups of A. Since A is quasinilpotent, we
+see that every element of A induces an inner automorphism on every minimal normal subgroup
+of A. Hence every element of A induces an inner automorphism on N.
+By analogy every
+element of B induces an inner automorphism on N.
+From G = AB it follows that every
+element of G induces an inner automorphism on N. So NCG(N) = G or G/CG(N) ≃ N. Now
+N ⋊ (G/CG(N)) ∈ (N∗)2. It means that N ≤ Z(N∗)2(G). Thus G ∈ (N∗)2 and h∗(G) ≤ 2, the
+final contradiction.
+Proof of Theorem 1.1(1). Let a group G be a mutually permutable product of subgroups A
+and B. From Theorem 2.6 and Lemma 4.3 it follows that
+max{h∗(A), h∗(B)} − 1 ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1.
+Assume that max{h∗(A), h∗(B)} − 1 = h∗(G). WLOG let h∗(A) = h∗(G) − 1. We may
+assume that a group G is the least order group with such properties. Then G has the unique
+minimal normal subgroup N, N ≤ A and h∗(A/N) = h∗(A) − 1 by Lemma 3.7.
+Assume that h∗(A/C∗
+A(N)) < h∗(A) − 1. Then
+F∗
+(h∗(A)−1)(A/N) = F∗
+(h∗(A)−1)(A)/N < A/N
+by Lemma 4.2. It means that h∗(A) = h∗(A/N), a contradiction. Hence h∗(A/C∗
+A(N)) =
+h∗(A) − 1.
+Since G/C∗
+G(N) = (AC∗
+G(N)/C∗
+G(N))(BC∗
+G(N)/C∗
+G(N)) is a mutually permutable products
+of subgroups AC∗
+G(N)/C∗
+G(N) and BC∗
+G(N)/C∗
+G(N) by (1) of Lemma 3.5 and A/C∗
+A(N) ≃
+AC∗
+G(N)/C∗
+A(N), we see that h∗(G/C∗
+G(N)) ≥ h∗(A/C∗
+A(N)) = h∗(A) − 1 by our assumptions.
+Note that F∗(G) ≤ C∗
+G(N). Now h∗(G)−1 = h∗(G/F∗(G)) ≥ h∗(G/C∗
+G(N)) ≥ h∗(A/C∗
+A(N)) =
+h∗(A) − 1. It means that h∗(G) ≥ h∗(A), the final contradiction.
+5
+Proof of Theorem 1.1(2)
+Lemma 5.1. Let p be a prime and H = Hp. If a group G = AB is a product of mutually
+permutable H-subgroups A and B, then G ∈ H.
+Proof. Assume the contrary. Let G be a minimal order counterexample.
+(1) G has a unique minimal normal subgroup N, G/N ∈ H and N is not p-soluble.
+Note that G/N is a mutually permutable product of H-subgroups (AN/N) and (BN/N) by
+(1) of Lemma 3.5. Hence G/N ∈ H by our assumption. Since H is a formation, we see that G
+8
+
+has a unique minimal normal subgroup. According to (4) of Lemma 3.5 AGBG ̸= 1. WLOG
+we may assume that G has a minimal normal subgroup N ≤ A.
+If N is p-soluble, then Fp(G)/N = Fp(G/N) = G, i.e. So Fp(G) = G. Thus G ∈ H, a
+contradiction.
+(2) N ≤ A ∩ B.
+Suppose that N ∩ B = 1. Note that N is not cyclic by (1). Then B ≤ CG(N) by (3) of
+Lemma 3.5. Hence N ⋊ G/CG(N) ≃ N ⋊ A/CA(N) ∈ H by [2, Corollary 2.2.5]. It means that
+N ≤ ZH(G). Therefore G ∈ H, a contradiction. Now N ∩ B ̸= 1. Hence N ≤ A ∩ B by (2) of
+Lemma 3.5.
+(4) The final contradiction.
+Since N is the unique minimal normal subgroup of G and non-abelian, we see that CG(N) =
+1. So CA(N) = CB(N) = 1. Hence Rp(A) = Rp(B) = 1. In particular F(A) = F(B) = 1.
+Note that all minimal normal subgroups of A are in N. For B is the same situation. Thus
+N = F∗(A) = F∗(B). So G/N is a mutually permutable product of p-soluble groups. Since the
+class of all p-soluble groups is closed by extensions by p-soluble groups, G/N is p-soluble by (1)
+and (4) of Lemma 3.5. From N ≤ F∗(G) it follows that G ∈ H, the contradiction.
+Proof of Theorem 1.1(2). Let H = Hp and a group G be a mutually permutable product of
+subgroups A and B. First we a going to prove that max{hH(A), hH(B)} = hH(G).
+By Lemmas 3.6 and 4.3 we have
+max{hH(A), hH(B)} − 1 ≤ hH(G) ≤ max{hH(A), hH(B)}.
+Assume that max{hH(A), hH(B)} − 1 = hH(G) for some mutually permutable product G of
+A and B. Assume that G is a minimal order group with this property. WLOG let hH(A) =
+hH(G) − 1. Then G has the unique minimal normal subgroup N, N ≤ A and hH(A/N) =
+hH(A) − 1 by Lemma 3.7.
+If N is p-soluble, then Rp(A/N) = Rp(A)/N. It means that Fp(A/N) = Fp(A)/N. Thus
+hH(A/N) = hH(A), a contradiction.
+It means that Rp(G) = 1. Note that now N is a simple non-abelian group. Since N is a
+unique minimal normal subgroup of G, we see that N = F∗(G). Now hH(G/N) = hH(G) − 1.
+Therefore
+hH(G) − 1 = hH(G/N) ≥ hH(A/N) = hH(A) − 1.
+Thus hH(G) ≥ hH(A), the contradiction.
+We proved that max{hH(A), hH(B)} = hH(G).
+Let G be a mutually permutable product of groups A and B. If A, B are p-soluble, then
+G is p-soluble by (1) and (4) of Lemma 3.5.
+Hence λp(G) = λp(A) = λp(B) = 0.
+Now
+assume that at least one of subgroups A, B is not p-soluble. Then G is not p-soluble by (1)
+and (4) of Lemma 3.5. WLOG let hH(A) ≥ hH(B). Hence A is not p-soluble. We proved
+that hH(A) = hH(G). Note that hH(G) = λp(G), hH(A) = λp(A), hH(B) = λp(B) if B is not
+p-soluble by Lemma 2.5 and 0 = λp(B) < 1 = hH(B) ≤ hH(A) = λp(A) otherwise. Thus
+max{λp(A), λp(B)} = λp(G).
+6
+Non-Frattini length
+The Frattini subgroup Φ(G) play an important role in the theory of classes of groups. One
+of the useful properties of the Fitting subgroup of a soluble group is that it is strictly greater
+than the Frattini subgroup of the same group. Note that the generalized Fitting subgroup is
+non-trivial in every group but there are groups in which it coincides with the Frattini subgroup.
+That is why the following length seems interesting.
+9
+
+Definition 6.1. Let 1 = G0 ≤ G1 ≤ · · · ≤ G2h = G be a shortest normal series in which for i
+even Gi+1/Gi ≤ Φ(G/Gi), and for i odd the factor Gi+1/Gi is a (non-empty) direct product of
+simple groups. Then h = ˜h(G) will be called the non-Frattini length of a group G.
+Note that if G is a soluble group, then ˜h(G) = h(G). Another reason that leads us to this
+length is the generalization of the Fitting subgroup ˜F(G) introduced by P. Schmid [16] and
+L.A. Shemetkov [17, Definition 7.5] and defined by
+Φ(G) ⊆ ˜F(G) and ˜F(G)/Φ(G) = Soc(G/Φ(G)).
+P. F¨orster [4] showed that ˜F(G) can be defined by
+Φ(G) ⊆ ˜F(G) and ˜F(G)/Φ(G) = F∗(G/Φ(G)).
+Let Φ and ˜F be functorials that assign Φ(G) and ˜F(G) to every group G. Then ˜F = Φ ⋆ F∗. It
+is well known that Φ satisfies (F1) and (F2). Hence ˜F satisfies (F1) and (F2) by Proposition
+2.2.
+Note that Φ(G/Φ(G)) ≃ 1. By analogy with the proof of Lemma 2.5 one can show that the
+non-Frattini length ˜h(G) of a group G and h˜F(G) coincide for every group G. The following
+theorem shows connections between the non-Frattini length and the generalized Fitting height.
+Theorem 6.1. For any group G holds ˜h(G) ≤ h∗(G) ≤ 2˜h(G). There exists a group H with
+˜h(H) = n and h∗(H) = 2n for any natural n.
+Proof. Since Φ(G) and Soc(G/Φ(G)) are quasinilpotent, we see that F∗(G) ≤ ˜F(G) ≤ F∗
+(2)(G).
+Now F∗
+(n)(G) ≤ ˜F(n)(G) ≤ F∗
+(2n)(G). Hence if ˜F(n)(G) = G, then F∗
+(n)(G) ≤ G and F∗
+(2n)(G) = G.
+It means ˜h(G) ≤ h∗(G) ≤ 2˜h(G).
+Let K be a group, K1 be isomorphic to the regular wreath product of A5 and K. Note
+that the base B of it is the unique minimal normal subgroup of K1 and non-abelian. According
+to [6], there is a Frattini F3K1-module A which is faithful for K1 and a Frattini extension
+A ֌ K2 ։ K1 such that A
+K1
+≃ Φ(K2) and K2/Φ(K2) ≃ K1.
+Let denote K2 by f(K). Now f(K)/˜F(f(K)) ≃ K. From the definition of h˜F = ˜h it follows
+that ˜h(f(K)) = ˜h(K) + 1.
+Note that Φ(f(K)) ⊆ F∗(f(K)).
+Assume that Φ(f(K)) ̸= F∗(f(K)).
+It means that
+F∗(f(K)) = ˜F(f(K)) is quasinilpotent. By [9, X, Theorem 13.8] it follows that Φ(f(K)) ⊆
+Z(F∗(f(K))). It means that 1 < B ≤ CK1(A). Thus A is not faithful, a contradiction.
+Thus Φ(f(K)) = F∗(f(K)) and f(K)/F∗(f(K)) ≃ K1.
+Since K1 has a unique mini-
+mal normal subgroup B and it is non-abelian, we see that F∗(K1) = B.
+It means that
+f(K)/F∗
+(2)(f(K)) ≃ K. From the definition of h∗ it follows that h∗(f(K)) = h∗(K) + 2.
+As usual, let f(1)(K) = f(K) and f(i+1)(K) = f(f(i)(K)).
+Then ˜h(f(n)(1)) = n and
+h∗(f(n)(1)) = 2n for any natural n.
+The following proposition directly follows from Theorem 2.6.
+Proposition 6.2. Let a group G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.
+Then ˜h(G) ≤ max{˜h(Ai) | 1 ≤ i ≤ n}.
+One of the main differences between the non-Frattini length and the generalized Fitting
+height is that the non-Frattini length of a normal subgroup can be greater than the non-Frattini
+length of a group.
+Example 6.1. Let E ≃ A5. There is an F5E-module V such that R = Rad(V ) is a faithful
+irreducible F5E-module and V/R is an irreducible trivial F5E-module (how to construct such
+module, for example, see [14]). Let G = V ⋋ E. Now Φ(G) = R by [3, B, Lemma 3.14]. Note
+10
+
+that G/Φ(G) = G/R ≃ Z5 × E. So ˜F(G) = G and ˜h(G) = 1. Note that G = V (RE) where V
+and RE are normal subgroups of G. Since V is abelian, we see that ˜h(V ) = 1. Note that R
+is a unique minimal normal subgroup of RE and Φ(RE) = 1. It means that ˜F(RE) = R and
+˜h(RE) = 2. Thus ˜h(G) < max{˜h(V ), ˜h(RE)} and ˜F does not satisfy (F3).
+Recall [1, Definition 4.1.1] that a group G is called a totally permutable product of its
+subgroups A and B if G = AB and every subgroup of A permutes with every subgroup of B.
+Theorem 6.3. Let a group G = AB be a totally permutable product of subgroups A and B.
+Then
+max{˜h(A), ˜h(B)} − 1 ≤ ˜h(G) ≤ max{˜h(A), ˜h(B)} + 1.
+Proof. If A = 1 or B = 1, then max{˜h(A), ˜h(B)} = ˜h(G). Assume that A, B ̸= 1.
+According to [1, Proposition 4.1.16] A ∩ B ≤ F(G). Hence A ∩ B ≤ F∗(G). Now G =
+G/F∗(G) is a totally permutable product of A = AF∗(G)/F∗(G) and B = BF∗(G)/F∗(G)
+by [1, Corollary 4.1.11]. Note that A ∩ B ≃ 1. According to [1, Lemma 4.2.2] [A, B] ≤ F(G).
+So [A, B] ≤ F∗(G). It means that
+G/F∗(G) = (AF∗(G)/F∗(G)) × (BF∗(G)/F∗(G)).
+Note that for the formation U of all supersoluble groups we have U ⊂ N2 ⊂ (N∗)2. Hence
+if H = H1H2 is a product of totally permutable (N∗)2-subgroups H1 and H2, then H ∈ (N∗)2
+by [1, Theorem 5.2.1]. Analyzing the proof of [1, Theorem 5.2.2] we see that this theorem is
+true not only for saturated formation, but for formations F = (G | G = ZF(G)). In particular,
+it is true for (N∗)2. Thus if H = H1H2 ∈ (N∗)2 is a product of totally permutable subgroups
+H1 and H2, then H1, H2 ∈ (N∗)2. Now (N∗)2 satisfies conditions of [1, Proposition 5.3.9].
+Therefore A ∩ F∗
+(2)(G) = F∗
+(2)(A) and B ∩ F∗
+(2)(G) = F∗
+(2)(B). Note that
+AF∗(G)/F∗(G) ≃ AF∗
+(2)(G)/F∗
+(2)(G) ≃ A/F∗
+(2)(A).
+By analogy BF∗(G)/F∗(G) ≃ B/F∗
+(2)(B). Hence
+G/F∗
+(2)(G) ≃ (A/F∗
+(2)(A)) × (B/F∗
+(2)(B)).
+By Theorem 2.6 and ˜h = h˜F we have ˜h(G/F∗
+(2)(G)) = max{˜h(A/F∗
+(2)(A)), ˜h(B/F∗
+(2)(B))}.
+From ˜F(H) ≤ F∗
+(2)(H) ≤ ˜F(2)(H) and Lemma 2.4 it follows that for any group H ̸= 1 holds
+˜h(H) − 1 = ˜h(H/˜F(H)) ≥ ˜h(H/F∗
+(2)(H)) ≥ ˜h(H/˜F(2)(H)) ≥ ˜h(H) − 2.
+Therefore
+{˜h(G) − ˜h(G/F∗
+(2)(G)), ˜h(A) − ˜h(A/F∗
+(2)(A)), ˜h(B) − ˜h(B/F∗
+(2)(B))} ⊆ {1, 2}.
+Thus max{˜h(A), ˜h(B)} − 1 ≤ ˜h(G) ≤ max{˜h(A), ˜h(B)} + 1.
+While proving Theorem 6.3 we were not able to answer the following question:
+Question 6.1. Let a group G = AB be a totally permutable product of subgroups A and B. Is
+max{˜h(A), ˜h(B)} ≤ ˜h(G)?
+The following question seems interesting
+Question 6.2. Do there exists a constant h with | max{˜h(A), ˜h(B)} − ˜h(G)| ≤ h for any
+mutually permutable product G = AB of subgroups A and B?
+D.A. Towers [19] defined and studied analogues of F∗(G) and ˜F(G) for Lie algebras. Using
+these subgroups and the radical (of a Lie algebra) one can introduce the generalized Fitting
+height, the non-soluble length and the non-Frattini length of a (finite dimension) Lie algebra.
+Question 6.3. Estimate the generalized Fitting height, the non-soluble length and the non-
+Frattini length of a (finite dimension) Lie algebra that is the sum of its two subalgebras (ideals,
+subideals, mutually or totally permutable subalgebras).
+11
+
+References
+[1] A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad. Products of Finite Groups. De
+Gruyter, 2010.
+[2] A. Ballester-Bollinches and L. M. Ezquerro. Classes of Finite Groups, volume 584 of Math.
+Appl. Springer Netherlands, 2006.
+[3] K. Doerk and T. O. Hawkes. Finite Soluble Groups, volume 4 of De Gruyter Exp. Math.
+De Gruyter, Berlin, New York, 1992.
+[4] P. F¨orster. Projektive Klassen endlicher Gruppen: IIa. Ges¨attigte Formationen: Ein all-
+gemeiner Satz von Gasch¨utz-Lubeseder-Baer-Typ. Pub, Mat. UAB, 29(2/3):39–76, 1985.
+[5] B. J. Gardner and R. Wiegandt. Radical theory of rings. Marcel Dekker New York, 2003.
+[6] R. L. Griess and P. Schmid. The Frattini module. Arch. Math., 30(1):256–266, 1978.
+[7] W. Guo. Structure Theory for Canonical Classes of Finite Groups. Springer-Verlag, Berlin,
+Heidelberg, 2015.
+[8] P. Hall and G. Higman. On the p-Length of p-Soluble Groups and Reduction Theorems
+for Burnside’s Problem. Proc. London Math. Soc., s3-6(1):1–42, 1956.
+[9] B. Huppert and N. Blackburn. Finite groups III, volume 243 of Grundlehren Math. Wiss.
+Springer-Verlag, Berlin, Heidelberg, 1982.
+[10] E. Jabara. The Fitting length of a product of mutually permutable finite groups. Acta
+Math. Hung., 159(1):206–210, 2019.
+[11] E. I. Khukhro and P. Shumyatsky. Nonsoluble and non-p-soluble length of finite groups.
+Isr. J. Math., 207(2):507–525, 2015.
+[12] E. I. Khukhro and P. Shumyatsky. On the length of finite factorized groups. Ann. Mat.
+Pura Appl., 194(6):1775–1780, 2015.
+[13] E. I. Khukhro and P. Shumyatsky. On the length of finite groups and of fixed points. Proc.
+Amer. Math. Soc., 143(9):3781–3790, 2015.
+[14] V.I. Murashka.
+On one conjecture about supersoluble groups.
+Publ. Math. Debrecen,
+100(3-4):399–404, 2022.
+[15] B. I. Plotkin. Radicals in groups, operations on group classes and radical classes. In Selected
+questions of algebra and logic, pages 205–244. Nauka, Novosibirsk, 1973. In Russian.
+[16] P. Schmid. ¨Uber die Automorphismengruppen endlicher Gruppen. Arch. Math., 23(1):236–
+242, 1972.
+[17] L. A. Shemetkov. Formations of finite groups. Nauka, Moscow, 1978. In Russian.
+[18] L. A. Shemetkov and A. N. Skiba. Formations of algebraic systems. Nauka, Moscow, 1989.
+In Russian.
+[19] D. A. Towers. The generalised nilradical of a Lie algebra. J. Algebra, 470:197–218, 2017.
+[20] J. S. Wilson. On the structure of compact torsion groups. Monatsh. Math., 96(1):57–66,
+1983.
+12
+

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+Secure synchronization of artificial neural networks
+used to correct errors in quantum cryptography
+Marcin Niemiec∗, Tymoteusz Widlarz∗, Miralem Mehic†‡
+∗ AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
+† Department of Telecommunications, Faculty of Electrical Engineering, University of Sarajevo,
+Zmaja od Bosne bb, 71000, Sarajevo, Bosnia and Herzegovina
+‡ VSB – Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czechia
+∗[email protected] †[email protected] ‡[email protected]
+Abstract—Quantum cryptography can provide a very high
+level of data security. However, a big challenge of this technique
+is errors in quantum channels. Therefore, error correction
+methods must be applied in real implementations. An example is
+error correction based on artificial neural networks. This paper
+considers the practical aspects of this recently proposed method
+and analyzes elements which influence security and efficiency.
+The synchronization process based on mutual learning processes
+is analyzed in detail. The results allowed us to determine the
+impact of various parameters. Additionally, the paper describes
+the recommended number of iterations for different structures of
+artificial neural networks and various error rates. All this aims
+to support users in choosing a suitable configuration of neural
+networks used to correct errors in a secure and efficient way.
+Index Terms—quantum cryptography, key reconciliation, error
+correction, artificial neural networks
+I. INTRODUCTION
+The emergence and intensive development of the field of
+quantum computing has put many cryptography algorithms at
+risk. However, quantum physics also allows to achieve multi-
+ple cryptography tasks. One of the most popular is quantum
+key distribution [1]. Unfortunately, quantum communication
+is not perfect and additional solutions are required to correct
+any errors after the key distribution in the quantum channel.
+Artificial neural networks can be utilized to correct these errors
+[2]. It is a recently proposed solution which provides high
+level of security and efficiency comparing to other existing
+error correction methods.
+This paper analyzes the impact of different neural networks’
+parameters on the synchronization process. These parameters
+influence the number of iterations required as well as the
+security and efficiency of quantum cryptography. Therefore,
+it is important to know which neural network scheme should
+be chosen and which should be avoided. Additionally, the syn-
+chronization requires the number of iterations to be specified.
+Therefore, a recommended number of iterations for a particular
+multiple neural network’s scheme is provided.
+The paper is structured as follows. Related work is re-
+viewed in Section 2. Section 3 presents the basics of quantum
+cryptography, the architecture of the tree parity machine,
+and error correction using this structure of artificial neural
+networks. Analysis of synchronization parameters including
+the recommended number of iterations for typical keys and
+error rates is described in Section 4. Section 5 concludes the
+paper.
+II. RELATED WORK
+The first quantum key distribution (QKD) protocol, intro-
+duced in 1984 by Bennet and Brassard, is BB84 [3]. This
+scheme uses the polarization state of a single photon to
+transmit information. Since then, several other protocols have
+been presented. One of them is the E91 protocol introduced
+in 1991 by Ekerd [4]. It utilizes entangled pairs of photons
+in the QKD process. However, some errors usually appear
+during data exchange in the quantum channel. After the initial
+QKD, there is a specific step: quantum bit error rate (QBER)
+estimation based on the acquired keys. The QBER value is
+usually low [5]. It must to be lower than the chosen threshold
+used to detect the eavesdropper.
+Several methods of correcting error incurred in the quan-
+tum key distribution process have been developed. The first
+described method – BBBSS – was proposed in 1992 [6].
+However, the most popular is the Cascade key reconciliation
+protocol [7]. It is based on multiple random permutations.
+The Winnow protocol, based on the exchange of parity and
+Hamming codes, is another method of error correction in the
+raw key [8]. Its main improvement is the reduction of the
+required communication between both parties. The third most
+popular error reconciliation scheme is the low density parity
+check approach. It offers a significant reduction of exchanged
+information; however, it introduces more computation and
+memory costs than the Cascade and Winnow protocols [7].
+In 2019, another method of error correction in quantum
+cryptography was proposed by Niemiec in [2]. The solution
+uses mutual synchronization of two artificial neural networks
+(ANN) to correct the errors. The tree parity machine (TPM)
+is proposed as a neural network used in this approach. It is
+a well-known structure in cryptography – the synchronization
+of two TPMs can be used as a key exchange protocol. TPMs
+arXiv:2301.11440v1  [cs.CR]  26 Jan 2023
+
+cannot be used as a general method to correct a selected error
+because it is not possible to predict the final string of bits after
+the synchronization process. However, it is a desirable feature
+for shared keys which should be random strings of bits.
+III. QUANTUM CRYPTOGRAPHY SUPPORTED BY
+ARTIFICIAL NEURAL NETWORKS
+Symmetric cryptography uses a single key to encrypt and
+decrypt secret messages. Let’s assume that Alice and Bob, the
+two characters used in describing cryptography protocols, are
+using symmetric encryption. The goal is to send information
+from Alice to Bob in a way that provides confidentiality. To
+achieve this, Alice and Bob need to agree on a shared secret
+key. Alice encrypts confidential data using the previously
+chosen key and Bob decrypts it using the same key. The same
+key is applied to encrypt and decrypt the information, hence
+the name: symmetric-key encryption. It is worth mentioning
+only the one-time-pad symmetric scheme has been proven
+secure but it requires a key not smaller than the message being
+sent.
+In general, symmetric-key encryption algorithms – for ex-
+ample the Advanced Encryption Standard (AES) [9] – per-
+form better than asymmetric-key algorithms [10]. However,
+symmetric-key algorithms have an important disadvantage
+compared to asymmetric-key schemes. In the symmetric key
+encryption scheme, the key needs to be safely distributed
+or established between Alice and Bob [11]. The symmetric
+key can be exchanged in a number of ways, including via
+a trusted third party or by direct exchange between involved
+parties. However, both methods introduce some vulnerabili-
+ties, including passive scanning of network traffic. A method
+where the eavesdropper can be easily detected uses quantum
+mechanics to establish keys between Alice and Bob. It is called
+the quantum key distribution protocol.
+A. Quantum key distribution
+Quantum mechanics allows for secure key distribution1
+among network users. Two main principles are the core of
+the security of QKD: an unknown quantum state cannot be
+copied [12], and the quantum state cannot be estimated without
+disturbing it. One of the most popular QKD protocols which
+uses those principles is the BB84 scheme [3].
+The BB84 protocol uses photons with two polarization
+bases: rectilinear or diagonal. Alice encodes a string of bits
+using photons on a randomly chosen basis. After that, all the
+photons are sent through a quantum channel. Bob randomly
+chooses a basis for each photon to decode the binary 0 or
+1. Alice and Bob’s bases are compared through a public
+communication channel. Each bit where both parties chose the
+same basis should be the same. However, when Bob measures
+the photon in a different basis than Alice, this bit is rejected.
+The remaining bits are the same for both parties and can be
+considered as a symmetric key. Next, the error estimation
+1In fact, a key is not distributed but negotiated. However, the term
+’distribution’ is consistently used in this paper to be consistent with the
+commonly accepted name of the technique.
+is performed. Randomly chosen parts of the keys between
+Alice and Bob are compared to compute the QBER value.
+If the comparison results in a high error rate, it means that
+the eavesdropper (Eve) is trying to gain information about
+the exchanged photons. However, the quantum channel is not
+perfect, and errors are usually detected due to disturbance,
+noise in the detectors or other elements. The number of errors
+introduced by the quantum channel’s imperfections must be
+considered while deciding the maximum acceptable error rate.
+The differences between Alice and Bob’s keys need to
+be corrected. Several error correction methods are known.
+BBBSS is the earliest scheme proposed in [6]. It is mainly
+based on parity checks. The most popular method is the
+Cascade protocol [13]. It is an improved version of BBBSS
+and requires less information to be sent between Alice and
+Bob through the public channel. The Cascade protocol and
+its predecessor are based on multiple parity checks. The basic
+idea is that the keys are divided into blocks of a fixed size.
+The number of bits in each block depends on the previously
+calculated QBER value. Alice and Bob compare the parities
+of each block to allow them to find an odd number of errors.
+If errors are detected in a given block, it is split into two.
+The process is repeated recursively for each block until all
+errors are corrected. It concludes a single iteration after which
+Alice and Bob have keys with an even number of errors or
+without any errors. Before performing the following iterations,
+the keys are scrambled, and the size of the block is increased.
+The number of iterations is predetermined. As a result of this
+process, Alice and Bob should have the same keys. However,
+it is not always the case. A number of iterations or block sizes
+can be chosen incorrectly and cause failure in error correction.
+Additionally, the algorithm performs multiple parity checks
+over the public channel, which can be intercepted by an
+eavesdropper (Eve). As a result, Eve can construct a partial
+key. Alice and Bob should discard parts of their keys to
+increase the lost security. This reduces the performance of
+this method since the confidential keys must be shortened in
+the process. Another error reconciliation method is based on
+mutual synchronization of artificial neural networks.
+B. Tree parity machine
+An artificial neural network (ANN) is a computing system
+inspired by biological neural networks [14]. ANNs are used
+to recognize patterns and in many other solutions in the fields
+of machine learning. ANNs consist of multiple connected
+nodes (artificial neurons), with each neuron representing a
+mathematical function [15]. These nodes are divided into three
+types of layers: the first (input) layer, at least one hidden layer,
+and the output layer. The connections between neurons in each
+layer can be characterized by weights.
+In cryptography, the most commonly used neural network is
+the tree parity machine (TPM) [16]. A scheme of this model
+is presented in Fig. 1. There are K ×N input neurons, divided
+into K groups. There is a single hidden layer with K nodes.
+Each of these nodes has N inputs. The TPM has a single
+output neuron. The connections between input neurons and
+
+hidden layer neurons are described by weights W – integers
+in the range [−L, L], thus L is the maximum and −L is
+the minimum weight value. The values of σ characterize the
+connections between the hidden layer neurons and an output
+neuron. The output value of the TPM is described by τ.
+The value of σ is calculated using the following formulas:
+σk = sgn(
+N
+�
+n=1
+xkn ∗ wkn)
+(1)
+sgn(z) =
+�
+−1
+z ≤ 0
+1
+z > 0
+(2)
+Due to the usage of the presented signum function, σ can take
+two values: 1 or −1. The output value of TPM is calculated
+as:
+τ =
+K
+�
+k=1
+σk
+(3)
+This neural network has two possible outcomes: 1 or −1.
+For the TPM structure, multiple learning algorithms are
+proposed. Most popular are Hebbian, anti-Hebbian, and ran-
+dom walk. The leading is the Hebbian rule [17]. The Hebbian
+algorithm updates ANN weights in the following manner:
+w∗
+kn = vL(wkn + xkn ∗ σk ∗ θ(σk, τ))
+(4)
+where θ limits the impact of hidden layer neurons whose value
+was different than τ:
+θ(σk, τ) =
+�
+0
+if σk ̸= τ
+1
+if σk = τ
+(5)
+The vL function makes sure that the new weights are kept
+within the [−L, L] range:
+vL(z) =
+�
+�
+�
+�
+�
+−L
+if z ≤ −L
+z
+if − L < z < L
+L
+if z ≥ L
+(6)
+The TPM structure allows for mutual learning of the two
+neural networks [18], primarily based on updating weights
+only when the outputs from both neural networks are the same.
+The input values are random and the same for both Alice and
+Bob’s TPMs. Inputs are updated in each iteration. The security
+of this process relies on the fact that cooperating TPMs can
+achieve convergence significantly faster than Eve’s machine,
+which can update weights less frequently. The TPM is most
+commonly used in cryptography to exchange a secret key. This
+usage is defined as neural cryptography [19]. Alice and Bob
+mutually synchronize their TPMs to achieve the same weights.
+After the synchronization process, these weights provide a
+secure symmetric key.
+C. Error correction based on TPMs
+TPMs can be utilized during the error correction process
+in quantum cryptography [2]. The neural network’s task is to
+correct all errors to achieve the same string of confidential bits
+at both endpoints. Firstly, Alice and Bob prepare their TPMs.
+The number of neurons in the hidden layer (K) and the number
+of input neurons (N) is determined by Alice and passed on
+to Bob. The value L must also be agreed between the users.
+The keys achieved using the QKD protocol are changed into
+integer values in the range [−L, L]. These values are used
+in the appropriate TPMs as weights between neurons in the
+input layer and the hidden layer. Since Alice’s string of bits
+is similar to Bob’s (QBER is usually not high), the weights
+in the created TPMs are almost synchronized. At this point,
+Alice and Bob have constructed TPMs with the same structure
+but with a few differences in the weight values.
+After establishing the TPM structure and changing bits to
+weights, the synchronization process starts. It consists of mul-
+tiple iterations, repeated until common weights are achieved
+between Alice and Bob. A single iteration starts from Alice
+choosing the input string and computing the result using the
+TPM. After that, the generated input string is passed on to Bob,
+who computes the output of his TPM using the received input.
+Then, the results are compared. If the outputs of both TPMs
+match, the weights can be updated. Otherwise, the process is
+repeated with a different input string.
+After an appropriate number of iterations, the TPMs are
+synchronized and Alice and Bob can change the weights back
+into a string of bits. The resulting bits are the same. However,
+the privacy amplification process after error correction is still
+recommended [20]. The reduction of the key protecting Alice
+and Bob from information leakage is defined as [2]:
+Z = log2L+12i
+(7)
+where i is the number of TPM iterations.
+This usage of TPM is safer than the neural cryptography
+solution, because weights are similar before the synchroniza-
+tion. Therefore, significantly fewer iterations are required to
+achieve convergence than the randomly initialized weights
+in key establishing algorithms. It is worth mentioning this
+method of error correction is characterized by high efficiency,
+e.g. requires approximately 30% less iterations than Cascade
+algorithm [2].
+IV. ANALYSIS OF THE SYNCHRONIZATION PROCESS
+The crucial decision regarding the error detection approach
+based on TPMs is the number of iterations during the syn-
+chronization process. This value should be as low as possible
+for security reasons. However, it cannot be too low, since
+neural networks will not be able to correct all errors in the
+key otherwise. It is the user’s responsibility to select the
+appropriate value for the error correction. The main objective
+of the analysis is to determine the impact of various neural
+network parameters on the synchronization process. Another
+goal is to provide a recommended number of iterations for
+users.
+
+X11
+X12
+X13
+X1N
+X21
+X22
+X23
+X2N
+XK1
+XK2
+XK3
+XKN
+∑
+∏
+∑
+∑
+W11
+W1N
+W21
+W2N
+WKN= {-L, … ,L}
+WK1
+σK= {-1, 1}
+σ2
+σ1
+τ={-1, 1}
+Fig. 1. Model of tree parity machine.
+A. Testbed
+The experiments require an application to simulate the error
+correction process based on artificial neural networks. The
+application for correcting errors arising in quantum key distri-
+bution was written in Python and uses the NumPy package – a
+library for scientific computing which provides fast operations
+on arrays required by the TPM. The functions provided by
+NumPy satisfy all necessary calculations to achieve neural
+network convergence. Synchronization of TPMs is performed
+over sockets to allow real-world usage of this tool. The
+Hebbian learning algorithm for updating weights is used.
+The developed application makes it possible to correct errors
+in the keys using quantum key distribution protocols. The users
+are also able to correct simulated keys with the chosen error
+rate. It helps if users do not have strings of bits created by a
+real QKD system. An important feature of the tool is its ability
+to select neural network parameters. The user can personalize
+the synchronization process, starting from the key length and
+error rate. The least sufficient number of bits was used for
+translation into a single integer (values of the weights must be
+in the range [−L, L]). It was demonstrated that the number of
+hidden neurons and the number of inputs depend on the chosen
+key length and L value. Therefore, users need to select these
+parameters taking into account the requirements and needs.
+During the experiments the minimum number of returned
+required iterations for a single TPM configuration was set
+to 200. The maximum number of iterations was limited to
+1000. Additionally, the maximum number of retries in a single
+iteration was limited to 10 to speed up the simulation process.
+Finally, 1880 different scenarios were analyzed. All possible
+TPM configurations for key lengths varying between 100 and
+700 with a 100 bit step are available. Moreover, the data is
+available for other keys with lengths varying between 128 and
+352 with an 8 bit step. Between 350 and 500 synchronizations
+were performed for each TPM. It was assumed that this
+number of iterations is sufficient to achieve convergence.
+B. Recommended number of iterations
+To obtain the recommended number of iterations of TPMs
+for successful error correction, the sum of means and standard
+deviations of the results was calculated. The median and
+variance values were calculated as well for comparison. The
+full results are available online2. The selected part – the neural
+network configurations where the key length equals 256 bits
+with the recommended number of iterations – is presented in
+Tab. I.
+Fig. 2. Histogram for number of iterations (TPM with a 256 bit key, N = 16,
+K = 4, L = 4, QBER = 3%).
+2Recommended numbers of iterations for 1880 different scenarios –
+TPM structures and QBER values – are available from: http://kt.agh.edu.pl/
+∼niemiec/ICC-2023 This is mainly based on possible key lengths which vary
+between 128 and 500 bits with 4 bit steps. Additionally, keys with lengths
+between 500 and 700 with 100 bit steps are included.
+
+180
+160
+140
+120
+Count
+100
+80
+60
+40
+20
+0
+[11, 69]
+(69, 127)
+(127, 185)
+(185, 243)
+(243, 301)
+(301, 359)
+(359, 400)
+> 400
+Number of iterationsTABLE I
+RECOMMENDED NUMBER OF ITERATIONS FOR TPMS GENERATED FOR
+256 BIT KEYS
+Weights
+range
+{−L, L}
+QBER
+[%]
+Number
+of
+inputs
+to a single
+hidden
+neuron
+[N]
+Number
+of
+hidden
+neurons
+[K]
+Recommended
+number
+of
+iterations
+2
+1
+2
+43
+154
+2
+1
+43
+2
+51
+2
+2
+2
+43
+179
+2
+2
+43
+2
+59
+2
+2
+86
+1
+24
+2
+3
+2
+43
+188
+2
+3
+43
+2
+64
+2
+3
+86
+1
+25
+3
+1
+2
+43
+218
+3
+1
+43
+2
+71
+3
+1
+86
+1
+33
+3
+2
+2
+43
+309
+3
+2
+43
+2
+94
+3
+2
+86
+1
+39
+3
+3
+2
+43
+325
+3
+3
+43
+2
+97
+3
+3
+86
+1
+40
+4
+1
+2
+32
+450
+4
+1
+4
+16
+496
+4
+1
+8
+8
+301
+4
+1
+16
+4
+176
+4
+1
+32
+2
+125
+4
+2
+2
+32
+554
+4
+2
+4
+16
+701
+4
+2
+8
+8
+483
+4
+2
+16
+4
+264
+4
+2
+32
+2
+152
+4
+3
+2
+32
+609
+4
+3
+4
+16
+772
+4
+3
+8
+8
+542
+4
+3
+16
+4
+302
+4
+3
+32
+2
+164
+Fig. 2 shows the histogram of data gathered for a sin-
+gle neural network configuration. The distribution is right-
+skewed. The mean value is greater than the median. It is a
+common characteristic for other tested TPM configurations. If
+the distribution is not positively skewed, it is symmetrical.
+The recommended number of iterations for the presented
+configuration, according to Tab. I, equals 302. It is based on
+the sum of the mean and standard deviation values. For all
+presented TPM configurations, this sum gives an 84% chance
+of successful synchronization, assuming a normal distribution
+of results. For the right-skewed distribution, similar to the one
+presented in Fig. 2, the probability of success is higher. The
+85-th percentile for the given set is equal to 276 – less than
+the proposed value. In this case, after choosing the suggested
+number of iterations the user has more than an 88% chance
+of success.
+Knowing the lowest required number of iterations is im-
+portant because it reduces the risk of a successful attack by
+Eve. The attacker could create independent TPMs and try
+to synchronize one of them with Alice or Bob’s machine.
+The recommended number of iterations increases the security
+of this solution because Alice and Bob require far fewer
+iterations to synchronize, compared to Alice (or Bob) and Eve
+synchronizing using random weights.
+C. Impact of TPM structures
+The results of simulations allow us to analyze how TPM
+structures affect the number of required iterations during the
+synchronization process. Fig. 3 shows the number of required
+iterations depending on the K and N parameters. It shows
+two different TPM configurations: one with a 144 bit key and
+another with a 216 bit key. These configurations were chosen
+due to having a similar number of possible K and N pairs.
+For a given key length, L value and error rate there is a limited
+number of possible N and K values. The K value changes
+in inverse proportion to the N value. As presented in Fig.
+3 the speed of the TPM synchronization process depends on
+the neural network structure (N and K values). The number
+of required iterations increases alongside the higher number
+of neurons in the hidden layer (K). The trend is similar for
+both presented TPMs. After achieving a certain threshold,
+the number of recommended iterations increases slowly. The
+results fit the logarithmic trend line. It means that after a
+certain K value, increasing this parameter further does not
+affect the synchronization speed as much as under a certain
+threshold.
+Fig. 3. Number of iterations for TPMs with 144 and 216 bit keys for different
+K value.
+Other configurations of the selected TPMs were studied
+based on the increasing error rate of the keys. Two configura-
+tions with 128 and 256 bit keys were tested. The average of
+every possible configuration of the recommended number of it-
+erations was calculated for different QBER values. The results
+are presented in Fig. 4. This confirms that a greater number
+of errors results in a higher average number of recommended
+iterations. It confirms the applicability of TPMs to correct
+errors emerging in quantum key distribution, where the error
+rate should not be higher than a few percent. Therefore, the
+eavesdropper needs more iterations to synchronize its TPM.
+Additionally, it was verified that value L has an exponential
+impact on the average recommended number of iterations. The
+data was gathered using a similar approach to the study with
+
+180
+160
+Recommended number of iterations
+140
+120
+100
+144 bits
+80
+216 bits
+60
+40
+20
+0
+6
+8
+9
+10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
+Number of hidden layer neurons (K)Fig. 4.
+Number of iterations for TPMs with 128 and 256 bit keys depended
+on the QBER.
+the impact of QBER. The average recommended number of
+iterations of each configuration for a given L was calculated.
+Fig. 5 shows the exponential trend line. It is worth mentioning
+that the impact of L value on the synchronization time is
+significant.
+Fig. 5.
+Number of iterations for TPMs with 128 and 256 bit keys dependent
+on the L value.
+It is the user’s responsibility to choose the best possible
+configuration for a given key length and QBER value. The
+analysis shows that the L value should be chosen carefully
+since it exponentially affects the required number of iterations.
+Additionally, the choice of the K value should be made
+with caution due to its logarithmic impact on the number of
+iterations.
+V. SUMMARY
+The analysis of the TPM synchronization process used for
+error correction purposes was presented in this paper. It shows
+that the parameters of the TPM structure have an impact on
+the synchronization time and security of this error correction
+method. However, different parameters of artificial neural
+networks have different effects. Therefore, users should be
+aware of how to choose the configuration of neural networks
+used to correct errors in a secure and efficient way. One of
+the deciding factors which need to be selected is the number
+of iterations. The paper describes the recommended number
+of iterations for different TPM structures and QBER values
+to assist users in this step. The numbers recommended by the
+authors are as low as possible but with a high probability of
+successful synchronization to ensure secure and efficient error
+correction based on artificial neural networks.
+ACKNOWLEDGMENT
+This work was supported by the ECHO project which has
+received funding from the European Union’s Horizon 2020
+research and innovation programme under the grant agreement
+no. 830943.
+REFERENCES
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+
+300
+ iterations
+250
+I number of i
+200
+150
+●128 bits
+●256 bits
+100
+50
+0
+1
+2
+3
+QBER [%]400
+300
+200
+·128 bits
+·256 bits
+100
+0
+2
+3
+4
+7

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+arXiv:2301.13305v1  [math.CO]  30 Jan 2023
+Graph-Codes
+Noga Alon ∗
+Abstract
+The symmetric difference of two graphs G1, G2 on the same set of vertices [n] =
+{1, 2, . . . , n} is the graph on [n] whose set of edges are all edges that belong to exactly
+one of the two graphs G1, G2. Let H be a fixed graph with an even (positive) number
+of edges, and let DH(n) denote the maximum possible cardinality of a family of graphs
+on [n] containing no two members whose symmetric difference is a copy of H. Is it
+true that DH(n) = o(2(n
+2)) for any such H? We discuss this problem, compute the
+value of DH(n) up to a constant factor for stars and matchings, and discuss several
+variants of the problem including ones that have been considered in earlier work.
+1
+Introduction
+1.1
+The problem
+The symmetric difference of two graph G1 = (V, E1) and G2 = (V, E2) on the same set of
+vertices V is the graph (V, E1 ⊕ E2) where E1 ⊕ E2 is the symmetric difference between
+E1 and E2, that is, the set of all edges that belong to exactly one of the two graphs. Put
+V = [n] = {1, 2, . . . , n} and let H be a family of graphs on the set of vertices [n] which is
+closed under isomorphism. A collection of graphs F on [n] is called an H-(graph)-code if
+it contains no two members whose symmetric difference is a graph in H. For the special
+case that H contains all copies of a single graph H on [n] this is called an H-code. Here
+we are interested in the maximum possible cardinality of such codes for various families
+H. Let DH(n) denote this maximum, and let
+dH(n) = DH(n)
+2(n
+2)
+denote the maximum possible fraction of the total number of graphs on [n] in an H-code.
+If H consists of all graphs isomorphic to one graph H, we denote dH(n) by dH(n). Note
+that if H consists of all graphs with less than d edges, then DH(n) is simply the maximum
+∗Princeton University, Princeton, NJ, USA and Tel Aviv University, Tel Aviv, Israel.
+Email:
+[email protected]. Research supported in part by NSF grant DMS-2154082 and by USA-Israel
+BSF grant 2018267.
+1
+
+possible cardinality of a binary code of length
+�n
+2
+�
+and minimum distance at least d. This
+motivates the terminology “graph-codes” used here.
+The case H = K where K is the family of all cliques is of particular interest. This case
+is motivated by a conjecture of Gowers raised in his blog post [8] in 2009 and is discussed
+briefly in the comments of that blog. If H consists of all graphs with independence number
+at most 2, then dH(n) ≥ 1/8 for all n ≥ 3, as shown by the family of all graphs on [n]
+containing a triangle on the set of vertices {1, 2, 3}. An interesting result of Ellis, Filmus
+and Friedgut [5], settling a conjecture of Simonovits and S´os, asserts that this is tight
+for all n ≥ 3. The corresponding result, that dH′(n) = 1/26 for all n ≥ 4, where H′ is
+the family of all graphs with independence number at most 3, is proved in [3]. A more
+systematic study of the parameters DH(n) and dH(n) for various families of graphs H
+appears in the recent paper [1]. The families H considered in this work include the family
+of all disconnected graphs, the family of all graphs that are not 2-connected, the family
+of all non-Hamiltonian graphs and the family of all graphs that contain or do not contain
+a spanning star. Additional families studied are all graphs that contain an induced or
+non-induced copy of a fixed graph T, or all graphs that do not contain such a subgraph.
+In this note we focus on the case that H consists of a single graph H and the case that
+H is the family of all cliques, or all cliques up to a prescribed size. Note that trivially,
+if every member of H has an odd number of edges then dH(n) ≥ 1
+2 as the family of all
+graphs on [n] with an even number of edges forms an H-code.
+This suggests the following intriguing question.
+Question 1.1. Let H be a family of graphs closed under isomorphism. Is it true that
+dH(n) tends to 0 as n tends to infinity if and only if H contains a graph with an even
+number of edges ? Equivalently: is it true that for any fixed graph H with an even number
+of edges, dH(n) tends to 0 as n tends to infinity ?
+We also study the linear variant of these problems, where the H-codes considered are
+restricted to linear subspaces, that is, to families of graphs on [n] closed under symmetric
+difference.
+1.2
+Results
+Recall that K is the family of all cliques. Let K(r) denote the set of all cliques on at most
+r vertices. Let K1,t denote the star with t edges and let Mt denote the matching of t edges.
+Theorem 1.2. For every positive integer k,
+dK1,2k(n) = Θk(1/nk) and
+dM1,2k(n) = Θk(1/nk).
+Proposition 1.3. For every integer r ≥ 1,
+dK(4r+3)(n) ≥ Ω( 1
+nr ).
+2
+
+Proposition 1.4. For the family K of all cliques, dK(n) ≥
+1
+2[n/2] .
+Proposition 1.5. Let H be a fixed graph obtained from two copies of a graph H′ by
+identifying the vertices of an independent set of H′. Then
+dH(n) ≤ |V (H)|
+n
+for all n ≥ |V (H)|.
+In particular, dH(n) tends to 0 as n tends to infinity.
+Remark:
+all lower bounds are proved by exhibiting proper colorings of the relevant
+Cayley graphs, and in all cases the constructed family is an affine space over Z2. Using
+a simple Ramsey-theoretic argument it is not difficult to show that for an affine space
+the maximum possible cardinality obtained is at most a fraction O(log log n/ log n) of all
+graphs on n vertices whenever the defining family contains a fixed graph with an even
+number of edges.
+Since all lower bounds are obtained by what may be called linear graph-codes one can
+study this separately, as done for standard error correcting codes. For the family of all
+cliques K we get here an exact result (strengthening the assertion of Proposition 1.4).
+Theorem 1.6. For any n ≥ 2, the minimum possible co-dimension of a linear space of
+graphs on n vertices that contains no member of K is exactly [n/2].
+2
+Proofs
+2.1
+Upper bounds
+For a family of graphs H and an integer n, the Cayley graph C(n, H) is the graph whose
+vertices are all graphs on the n vertices [n], where two are adjacent iff their symmetric
+difference is a member of H. This is clearly a Cayley graph over the elementary abelian
+2-group ZN
+2 with N =
+�n
+2
+�
+. The function DH(n) is just the independence number of this
+graph, dH(n) is the so called independence ratio. Since the graph C(n, H) is vertex tran-
+sitive, its independence ratio is exactly the reciprocal of its fractional chromatic number.
+In order to prove an upper bound of α for its independence ratio it suffices to exhibit a
+set S of vertices that contains no independent set of size larger than α|S|. This applies
+also to weighted sets of vertices, but we will not use weights here.
+Proof of Proposition 1.5: Let a + b denote the number of vertices of H′ where b is the
+size of its independent set so that H is obtained from two copies of H′ by identifying the
+vertices in this independent set. Thus the number of vertices of H is 2a + b. Consider
+the following set of m = ⌊(n − b)/a⌋ copies of H′ on subsets of the vertex set [n]. All of
+3
+
+them contain the same independent set on the vertices {n − b + 1, n − b + 2, . . . , n}, and
+the additional vertices of copy number i are the vertices (i − 1)a + 1, (i − 1)a + 2, . . . , ia},
+where 1 ≤ i ≤ m. Each of these copies can be viewed as a vertex of the Cayley graph
+C = C(n, {H}). Since the symmetric difference of every pair of such copies forms a copy
+of H, this set forms a clique of size m in C, implying that dH(n) ≤ 1
+m ≤ |V (H)|/n.
+□
+The proofs of Theorem 1.2 for stars and for matchings are very similar. We describe the
+proof for stars and briefly mention the modification needed for matchings. The upper
+bound in Theorem 1.2 for the star K1,1 is a special case of the result above (with H′ being
+a single edge). The upper bound for any prime k can be proved using the following result
+of Frankl and Wilson.
+Theorem 2.1 ([7]). Let p be a prime, and let a0, a1, . . . , ar be distinct residue classes
+modulo p. Let F be a family of subsets of [n] and suppose that |F| ≡ a0 mod p for all
+F ∈ F and that for every two distinct F1, F2 ∈ F, |F1∩F2| ≡ ai mod p for some 1 ≤ i ≤ r.
+Then |F| ≤ �r
+i=0
+�n
+i
+�
+.
+Suppose k is a prime, n ≥ 2k and consider the family G of all stars K1,2k−1 with
+center 1 and 2k − 1 leaves among the vertices {2, 3, . . . , n}. Thus |G| =
+� n−1
+2k−1
+�
+. If two
+such stars share exactly k − 1 common leaves then their symmetric difference is a copy of
+K1,2k. A subset of G which is independent in the Cayley graph C(n, K1,2k) corresponds to
+a collection of subsets of the set {2, 3, . . . , n}, each of size 2k − 1, where the intersection of
+no two of these subsets is of cardinality k−1. Therefore, each of these sets is of cardinality
+−1 modulo k and no intersection is of cardinality −1 modulo k. By the Frankl-Wilson
+Theorem (Theorem 2.1) the cardinality of such a family is at most �k−1
+i=0
+�n−1
+i
+�
+. Therefore,
+for every prime k,
+dK1,2k(n) ≤
+�k−1
+i=0
+�n−1
+i
+�
+� n−1
+2k−1
+�
+≤ Ok( 1
+nk ).
+In order to prove the upper bound for all k we need the following result of Frankl and
+F¨uredi.
+Theorem 2.2 ([6]). For every fixed positive integers ℓ > ℓ1 +ℓ2 there exist n0 = n0(ℓ) and
+dℓ > 0 so that for all n > n0, if F is a family of ℓ-subsets of [n] in which the intersection
+of each pair of distinct members is of cardinality either at least ℓ − ℓ1 or strictly smaller
+than ℓ2, then
+|F| ≤ dℓ · nmax{ℓ1,ℓ2}.
+Proof of Theorem 1.2, upper bound: The proof for stars is essentially identical to
+the one described above for prime k, using Theorem 2.2 instead of Theorem 2.1.
+Let
+G be the family of all stars K1,2k−1 with center 1 and 2k − 1 leaves among the vertices
+4
+
+{2, 3, . . . , n}. Thus |G| =
+� n−1
+2k−1
+�
+. If two such stars share exactly k − 1 common leaves
+then their symmetric difference is a copy of K1,2k. Therefore, by Theorem 2.2 above with
+ℓ = 2k−1, ℓ1 = ℓ2 = k−1, the maximum cardinality of a subset of G which is independent
+in the Cayley graph C(n, K1,2k) is at most some ck(n − 1)k−1 for all sufficiently large n.
+This supplies the required upper bound
+ck(n − 1)k
+|G|
+≤ Ok( 1
+nk ),
+for dK1,2k(n). The proof for matchings is similar, starting with the family of all subsets
+of cardinality 2k − 1 of a fixed matching of cardinality ⌊n/2⌋. The symmetric difference
+of any two matchings that share exactly k − 1 common edges is a copy of M2k. Thus the
+proof can proceed exactly as in the case of stars.
+□
+2.2
+Lower bounds
+In order to lower bound the independence number of a Cayley graph C = C(n, H) it
+suffices to upper bound its chromatic number. One way to do so is to assign to each edge
+e of the complete graph on [n] a vector ve ∈ Zr
+2 for some r, so that for every H ∈ H,
+�
+e∈E(H) ve ̸= 0, where the sum is computed in Zr
+2. Given these vectors, we can assign
+to each graph G on [n] the color �
+e∈E(G) ve (computed, of course, in Zr
+2). This is clearly
+a proper coloring of C by at most 2r colors. Note that the matrix whose columns are
+the
+�n
+2
+�
+vectors ve is the analogue of the parity-check matrix of a linear error correcting
+code in the traditional theory of codes, and the color defined above is the analogue of the
+syndrome of a word, see, e.g., [9] for more information about these basic notions.
+Proof of Theorem 1.2, lower bound: For stars, it suffices to show that the chromatic
+number of the Cayley graph C = C(n, K1,2k) is at most O(nk). Let s be the smallest
+integer so that 2s − 1 ≥ n. As shown by the columns of the parity check matrix of a
+BCH-code with designed distance 2k + 1 there is a collection S of 2s − 1 binary vectors
+of length r = ks so that no sum of at most 2k of them (in Zks
+2 ) is the zero vector. Fix a
+proper edge coloring c of Kn by n colors. For each edge e let ve be the vector number c(e)
+in S. This gives the desired lower bound for stars. For matchings we use essentially the
+same construction, starting with a (non-proper) edge coloring of Kn by n colors in which
+each color class forms a star.
+□
+Proof of Proposition 1.3, lower bound:
+As in the previous proof, but the initial
+edge-coloring now is defined by c(ij) = i for all i < j and the binary vectors selected
+are taken from the columns of the parity check matrix of a code with designed distance
+2r + 2. Let U be the set of vertices of a clique of size at least 2 and at most 4r + 3. Then
+U contains at least 1 and at most 2r + 1 vertices i for which there is an odd number of
+5
+
+vertices of U with index strictly larger than i. Therefore the sum of vectors corresponding
+to the edges of the clique on U is equal to a sum of at most 2r + 1 column vectors of the
+parity check matrix, which is nonzero.
+□
+Proof of Proposition 1.4, lower bound: This follows from the construction in the
+proof of Theorem 1.6 described in the next section.
+3
+Linear graph-codes
+Proof of Theorem 1.6: The theorem is equivalent to the statement that for all n ≥ 2
+the minimum possible r = r(n) so that there are graphs G1, . . . , Gr on the vertex set [n]
+such that every clique on a subset of cardinality at least 2 of [n] contains an odd number
+of edges of at least one graph Gi, is r = [n/2]. It clearly suffices to prove the upper bound
+for odd n (that imply the result for n − 1) and the lower bound for even n (implying the
+result for n + 1). The upper bound is described in what follows. Let n ≥ 3 be odd. Split
+the numbers [n − 1] = {1, 2, . . . , n − 1} into the (n − 1)/2 blocks Bi = {2i − 1, 2i} for
+1 ≤ i ≤ (n − 1)/2. Let Gi be the graph consisting of all edges of the n − 2i triangles with
+a common base Bi on the vertices Bi ∪ {j} for 2i < j ≤ n. Our family of graphs is the
+set of these (n − 1)/2 graphs Gi. Let K be an arbitrary clique on a subset A of at least
+2 vertices in [n]. If A contains a full block Bi for some i, then it contains exactly 2x + 1
+edges of Gi, where x is the cardinality of the intersection of A with {2i + 1, 2i + 2, . . . , n}.
+As this is odd for all x ≥ 0 we may assume that A contains no block Bi. In this case,
+let j be the second largest element in A (recall that |A| ≥ 2). Clearly j ≤ n − 1, hence
+it is contained in one of the blocks Bi. But in this case Gi contains exactly one edge
+of the clique K, completing the proof of the upper bound. Note that it is simple to give
+additional constructions with the same properties as any set of graphs that spans the same
+subspace as the graphs above will do. In particular, we can replace one of the graphs Gi
+by the complete graph Kn, which is the sum of all graphs Gi.
+To prove the lower bound assume n is even and let G1, . . . Gn/2−1 be a family of n/2−1
+graphs on [n]. We have to show that there is a clique on at least 2 vertices containing an
+even number of edges of each Gi. We show that in fact there is such a clique on an even
+number of vertices. To do so we apply the classical theorem of Chevalley and Warning
+(cf., e.g., [2] or [12]). Recall that it asserts that any system of polynomials with n variables
+over a finite field in which the number of variables exceeds the sum of the degrees, which
+admits a solution, must admit another one (in fact, the number of solutions is divisible by
+the characteristics). Associate each vertex i with a variable xi over Z2 and consider the
+following homogeneous system of polynomial equations over Z2. For each graph Gs in our
+6
+
+family,
+�
+ij∈E(Gs)
+xixj = 0.
+In addition, add the linear equation �n
+i=1 xi = 0.
+The sum of the degrees of the polynomials here is 2(n/2 − 1) + 1 = n − 1, which
+is smaller than the number of variables.
+Since the system is homogeneous it admits
+the trivial solution xi = 0 for all i. Any other solution (which exists by the Chevalley
+Warning Theorem) gives a clique on the set of vertices {i : xi = 1} which is nonempty, of
+even cardinality, and contains an even number of edges (possibly zero) of each Gi. This
+establishes the lower bound and completes the proof of Theorem 1.6.
+□
+4
+Concluding remarks and open problems
+• Question 1.1, which is equivalent to the problem of deciding whether or not for any
+fixed nonempty graph H with an even number of edges dH(n) tends to 0 as n tends
+to infinity, remains wide open.
+An interesting special case is whether or not dK4(n) = o(1). It is also interesting
+to decide whether or not dK4(n) ≥
+1
+no(1) . It is not difficult to show that the latter
+would follow from the existence (if true) of an edge coloring of Kn by no(1) colors
+with no copy of K4 in which every color appears an even number of times. This may
+be related to the construction in [10], see also [4].
+• Gowers conjectured in [8] that any family of a constant fraction of all graphs on [n],
+where n is sufficiently large, contains two graphs G1, G2 such that G2 is a subgraph
+of G1 and the symmetric difference of the two graphs (that is, the set of all edges of
+G1 that are not in G2) forms a clique. This is clearly stronger than the conjecture
+that dK(n) tends to 0 as n tends to infinity, which is also open. As explained in
+[8] the question of Gowers can be viewed as the first unknown case of a polynomial
+version of the density Hales-Jewett Theorem.
+• As mentioned in the remark following the statement of Proposition 1.5, it is not
+difficult to show that for every graph H with an even number of eges the maximum
+possible cardinality of a linear family of graphs on [n] in which no symmetric differ-
+ence is a copy of H, is o(2(n
+2)). As the proof applies Ramsey’s Theorem, it provides
+very weak bounds. It will be interesting to establish tighter bounds for the linear
+case. Theorem 1.6 provides an example of a tight result of this form.
+• The problem considered above can be extended to hypergraphs. More generally, it
+can be extended to other versions of problems about binary codes, where the coordi-
+nates of each codeword are indexed by the elements of some combinatorial structure,
+7
+
+and the forbidden symmetric differences correspond to a prescribed family of sub-
+structures. Here is an example of a problem of this type. What is the maximum
+possible cardinality of a collection of binary vectors whose coordinates are indexed
+by the elements of the ordered set [n], where no symmetric difference of two dis-
+tinct members of the collection forms an interval of length which is a cube of an
+integer? The corresponding Cayley graph here has 2n vertices, and it is triangle-free
+by Fermat’s last Theorem for cubes. Its independece number, which is the answer
+to the question above, is o(2n). Indeed, this follows from the Furstenberg-S´ark¨ozy
+Theorem and its extensions [11], by considering the maximum possible cardinality
+of an independent set in the induced subgraph on the set of all vertices that are
+characteristic vectors of an interval [i] = {1, . . . , i} for 0 ≤ i ≤ n.
+References
+[1] N. Alon, A. Gujgiczer, J. K¨orner, A. Milojevi´c and G. Simonyi, Structured codes of
+graphs, SIAM J. Discrete Math., to appear.
+[2] Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York,
+1966.
+[3] A. Berger and Y. Zhao, K4-intersecting families of graphs, arXiv:2103.12671, 2021.
+[4] D. Conlon, J. Fox, C. Lee and B. Sudakov, The Erd˝os-Gy´arf´as problem on generalized
+Ramsey numbers, Proc. London Math. Soc. 110 (2015), 1–18.
+[5] D. Ellis, Y. Filmus and E. Friedgut, Triangle-intersecting families of graphs, Journal
+of the European Mathematical Society 14 (2012), No. 3, 841–885.
+[6] P. Frankl and Z. F¨uredi, Forbidding just one intersection, J. Combin. Theory Ser. A
+39 (1985), no. 2, 160–176.
+[7] P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences,
+Combinatorica 1 (1981), 357–368.
+[8] W. T. Gowers, https://gowers.wordpress.com/2009/11/14/the-first-unknown-case-of-
+polynomial-dhj/
+[9] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, I. North-
+Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-
+New York-Oxford (1977).
+[10] D. Mubayi, Edge-coloring cliques with three colors on all 4-cliques, Combinatorica 18
+(1998), no. 2, 293–296.
+8
+
+[11] A. S´ark¨ozy, On difference sets of sequences of integers. III. Acta Math. Acad. Sci.
+Hungar. 31 (1978), no. 3-4, 355–386.
+[12] W. M. Schmidt, Equations over Finite Fields, an Elementary Approach, Springer
+Verlag Lecture Notes in Math., 1976.
+9
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='13305v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='CO]  30 Jan 2023  Graph-Codes  Noga Alon ∗  Abstract  The symmetric difference of two graphs G1, G2 on the same set of vertices [n] =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n} is the graph on [n] whose set of edges are all edges that belong to exactly  one of the two graphs G1, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let H be a fixed graph with an even (positive) number  of edges, and let DH(n) denote the maximum possible cardinality of a family of graphs  on [n] containing no two members whose symmetric difference is a copy of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Is it  true that DH(n) = o(2(n  2)) for any such H?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We discuss this problem, compute the  value of DH(n) up to a constant factor for stars and matchings, and discuss several  variants of the problem including ones that have been considered in earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1  The problem  The symmetric difference of two graph G1 = (V, E1) and G2 = (V, E2) on the same set of  vertices V is the graph (V, E1 ⊕ E2) where E1 ⊕ E2 is the symmetric difference between  E1 and E2, that is, the set of all edges that belong to exactly one of the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Put  V = [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n} and let H be a family of graphs on the set of vertices [n] which is  closed under isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A collection of graphs F on [n] is called an H-(graph)-code if  it contains no two members whose symmetric difference is a graph in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For the special  case that H contains all copies of a single graph H on [n] this is called an H-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Here  we are interested in the maximum possible cardinality of such codes for various families  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let DH(n) denote this maximum, and let  dH(n) = DH(n)  2(n  2)  denote the maximum possible fraction of the total number of graphs on [n] in an H-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  If H consists of all graphs isomorphic to one graph H, we denote dH(n) by dH(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note  that if H consists of all graphs with less than d edges, then DH(n) is simply the maximum  ∗Princeton University, Princeton, NJ, USA and Tel Aviv University, Tel Aviv, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Email:  nalon@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Research supported in part by NSF grant DMS-2154082 and by USA-Israel  BSF grant 2018267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  1  possible cardinality of a binary code of length  �n  2  �  and minimum distance at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This  motivates the terminology “graph-codes” used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  The case H = K where K is the family of all cliques is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This case  is motivated by a conjecture of Gowers raised in his blog post [8] in 2009 and is discussed  briefly in the comments of that blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If H consists of all graphs with independence number  at most 2, then dH(n) ≥ 1/8 for all n ≥ 3, as shown by the family of all graphs on [n]  containing a triangle on the set of vertices {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' An interesting result of Ellis, Filmus  and Friedgut [5], settling a conjecture of Simonovits and S´os, asserts that this is tight  for all n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The corresponding result, that dH′(n) = 1/26 for all n ≥ 4, where H′ is  the family of all graphs with independence number at most 3, is proved in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A more  systematic study of the parameters DH(n) and dH(n) for various families of graphs H  appears in the recent paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The families H considered in this work include the family  of all disconnected graphs, the family of all graphs that are not 2-connected, the family  of all non-Hamiltonian graphs and the family of all graphs that contain or do not contain  a spanning star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Additional families studied are all graphs that contain an induced or  non-induced copy of a fixed graph T, or all graphs that do not contain such a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In this note we focus on the case that H consists of a single graph H and the case that  H is the family of all cliques, or all cliques up to a prescribed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note that trivially,  if every member of H has an odd number of edges then dH(n) ≥ 1  2 as the family of all  graphs on [n] with an even number of edges forms an H-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  This suggests the following intriguing question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let H be a family of graphs closed under isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Is it true that  dH(n) tends to 0 as n tends to infinity if and only if H contains a graph with an even  number of edges ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Equivalently: is it true that for any fixed graph H with an even number  of edges, dH(n) tends to 0 as n tends to infinity ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  We also study the linear variant of these problems, where the H-codes considered are  restricted to linear subspaces, that is, to families of graphs on [n] closed under symmetric  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2  Results  Recall that K is the family of all cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let K(r) denote the set of all cliques on at most  r vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let K1,t denote the star with t edges and let Mt denote the matching of t edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For every positive integer k,  dK1,2k(n) = Θk(1/nk) and  dM1,2k(n) = Θk(1/nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For every integer r ≥ 1,  dK(4r+3)(n) ≥ Ω( 1  nr ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  2  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For the family K of all cliques, dK(n) ≥  1  2[n/2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let H be a fixed graph obtained from two copies of a graph H′ by  identifying the vertices of an independent set of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Then  dH(n) ≤ |V (H)|  n  for all n ≥ |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In particular, dH(n) tends to 0 as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Remark:  all lower bounds are proved by exhibiting proper colorings of the relevant  Cayley graphs, and in all cases the constructed family is an affine space over Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Using  a simple Ramsey-theoretic argument it is not difficult to show that for an affine space  the maximum possible cardinality obtained is at most a fraction O(log log n/ log n) of all  graphs on n vertices whenever the defining family contains a fixed graph with an even  number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Since all lower bounds are obtained by what may be called linear graph-codes one can  study this separately, as done for standard error correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For the family of all  cliques K we get here an exact result (strengthening the assertion of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For any n ≥ 2, the minimum possible co-dimension of a linear space of  graphs on n vertices that contains no member of K is exactly [n/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  2  Proofs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1  Upper bounds  For a family of graphs H and an integer n, the Cayley graph C(n, H) is the graph whose  vertices are all graphs on the n vertices [n], where two are adjacent iff their symmetric  difference is a member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This is clearly a Cayley graph over the elementary abelian  2-group ZN  2 with N =  �n  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The function DH(n) is just the independence number of this  graph, dH(n) is the so called independence ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Since the graph C(n, H) is vertex tran-  sitive, its independence ratio is exactly the reciprocal of its fractional chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In order to prove an upper bound of α for its independence ratio it suffices to exhibit a  set S of vertices that contains no independent set of size larger than α|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This applies  also to weighted sets of vertices, but we will not use weights here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='5: Let a + b denote the number of vertices of H′ where b is the  size of its independent set so that H is obtained from two copies of H′ by identifying the  vertices in this independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus the number of vertices of H is 2a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Consider  the following set of m = ⌊(n − b)/a⌋ copies of H′ on subsets of the vertex set [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' All of  3  them contain the same independent set on the vertices {n − b + 1, n − b + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}, and  the additional vertices of copy number i are the vertices (i − 1)a + 1, (i − 1)a + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , ia},  where 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Each of these copies can be viewed as a vertex of the Cayley graph  C = C(n, {H}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Since the symmetric difference of every pair of such copies forms a copy  of H, this set forms a clique of size m in C, implying that dH(n) ≤ 1  m ≤ |V (H)|/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  The proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 for stars and for matchings are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We describe the  proof for stars and briefly mention the modification needed for matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The upper  bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 for the star K1,1 is a special case of the result above (with H′ being  a single edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The upper bound for any prime k can be proved using the following result  of Frankl and Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let p be a prime, and let a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , ar be distinct residue classes  modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let F be a family of subsets of [n] and suppose that |F| ≡ a0 mod p for all  F ∈ F and that for every two distinct F1, F2 ∈ F, |F1∩F2| ≡ ai mod p for some 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Then |F| ≤ �r  i=0  �n  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Suppose k is a prime, n ≥ 2k and consider the family G of all stars K1,2k−1 with  center 1 and 2k − 1 leaves among the vertices {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus |G| =  � n−1  2k−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If two  such stars share exactly k − 1 common leaves then their symmetric difference is a copy of  K1,2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A subset of G which is independent in the Cayley graph C(n, K1,2k) corresponds to  a collection of subsets of the set {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}, each of size 2k − 1, where the intersection of  no two of these subsets is of cardinality k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore, each of these sets is of cardinality  −1 modulo k and no intersection is of cardinality −1 modulo k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' By the Frankl-Wilson  Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1) the cardinality of such a family is at most �k−1  i=0  �n−1  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore,  for every prime k,  dK1,2k(n) ≤  �k−1  i=0  �n−1  i  �  � n−1  2k−1  �  ≤ Ok( 1  nk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In order to prove the upper bound for all k we need the following result of Frankl and  F¨uredi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For every fixed positive integers ℓ > ℓ1 +ℓ2 there exist n0 = n0(ℓ) and  dℓ > 0 so that for all n > n0, if F is a family of ℓ-subsets of [n] in which the intersection  of each pair of distinct members is of cardinality either at least ℓ − ℓ1 or strictly smaller  than ℓ2, then  |F| ≤ dℓ · nmax{ℓ1,ℓ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2, upper bound: The proof for stars is essentially identical to  the one described above for prime k, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 instead of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Let  G be the family of all stars K1,2k−1 with center 1 and 2k − 1 leaves among the vertices  4  {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus |G| =  � n−1  2k−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If two such stars share exactly k − 1 common leaves  then their symmetric difference is a copy of K1,2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 above with  ℓ = 2k−1, ℓ1 = ℓ2 = k−1, the maximum cardinality of a subset of G which is independent  in the Cayley graph C(n, K1,2k) is at most some ck(n − 1)k−1 for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  This supplies the required upper bound  ck(n − 1)k  |G|  ≤ Ok( 1  nk ),  for dK1,2k(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The proof for matchings is similar, starting with the family of all subsets  of cardinality 2k − 1 of a fixed matching of cardinality ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The symmetric difference  of any two matchings that share exactly k − 1 common edges is a copy of M2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus the  proof can proceed exactly as in the case of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2  Lower bounds  In order to lower bound the independence number of a Cayley graph C = C(n, H) it  suffices to upper bound its chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' One way to do so is to assign to each edge  e of the complete graph on [n] a vector ve ∈ Zr  2 for some r, so that for every H ∈ H,  �  e∈E(H) ve ̸= 0, where the sum is computed in Zr  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Given these vectors, we can assign  to each graph G on [n] the color �  e∈E(G) ve (computed, of course, in Zr  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This is clearly  a proper coloring of C by at most 2r colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note that the matrix whose columns are  the  �n  2  �  vectors ve is the analogue of the parity-check matrix of a linear error correcting  code in the traditional theory of codes, and the color defined above is the analogue of the  syndrome of a word, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', [9] for more information about these basic notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2, lower bound: For stars, it suffices to show that the chromatic  number of the Cayley graph C = C(n, K1,2k) is at most O(nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let s be the smallest  integer so that 2s − 1 ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' As shown by the columns of the parity check matrix of a  BCH-code with designed distance 2k + 1 there is a collection S of 2s − 1 binary vectors  of length r = ks so that no sum of at most 2k of them (in Zks  2 ) is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Fix a  proper edge coloring c of Kn by n colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For each edge e let ve be the vector number c(e)  in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This gives the desired lower bound for stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For matchings we use essentially the  same construction, starting with a (non-proper) edge coloring of Kn by n colors in which  each color class forms a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='3, lower bound:  As in the previous proof, but the initial  edge-coloring now is defined by c(ij) = i for all i < j and the binary vectors selected  are taken from the columns of the parity check matrix of a code with designed distance  2r + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let U be the set of vertices of a clique of size at least 2 and at most 4r + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Then  U contains at least 1 and at most 2r + 1 vertices i for which there is an odd number of  5  vertices of U with index strictly larger than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore the sum of vectors corresponding  to the edges of the clique on U is equal to a sum of at most 2r + 1 column vectors of the  parity check matrix, which is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='4, lower bound: This follows from the construction in the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6 described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  3  Linear graph-codes  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6: The theorem is equivalent to the statement that for all n ≥ 2  the minimum possible r = r(n) so that there are graphs G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , Gr on the vertex set [n]  such that every clique on a subset of cardinality at least 2 of [n] contains an odd number  of edges of at least one graph Gi, is r = [n/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It clearly suffices to prove the upper bound  for odd n (that imply the result for n − 1) and the lower bound for even n (implying the  result for n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The upper bound is described in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let n ≥ 3 be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Split  the numbers [n − 1] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n − 1} into the (n − 1)/2 blocks Bi = {2i − 1, 2i} for  1 ≤ i ≤ (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let Gi be the graph consisting of all edges of the n − 2i triangles with  a common base Bi on the vertices Bi ∪ {j} for 2i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Our family of graphs is the  set of these (n − 1)/2 graphs Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let K be an arbitrary clique on a subset A of at least  2 vertices in [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If A contains a full block Bi for some i, then it contains exactly 2x + 1  edges of Gi, where x is the cardinality of the intersection of A with {2i + 1, 2i + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  As this is odd for all x ≥ 0 we may assume that A contains no block Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' In this case,  let j be the second largest element in A (recall that |A| ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Clearly j ≤ n − 1, hence  it is contained in one of the blocks Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' But in this case Gi contains exactly one edge  of the clique K, completing the proof of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note that it is simple to give  additional constructions with the same properties as any set of graphs that spans the same  subspace as the graphs above will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' In particular, we can replace one of the graphs Gi  by the complete graph Kn, which is the sum of all graphs Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  To prove the lower bound assume n is even and let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Gn/2−1 be a family of n/2−1  graphs on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We have to show that there is a clique on at least 2 vertices containing an  even number of edges of each Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We show that in fact there is such a clique on an even  number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' To do so we apply the classical theorem of Chevalley and Warning  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', [2] or [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Recall that it asserts that any system of polynomials with n variables  over a finite field in which the number of variables exceeds the sum of the degrees, which  admits a solution, must admit another one (in fact, the number of solutions is divisible by  the characteristics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Associate each vertex i with a variable xi over Z2 and consider the  following homogeneous system of polynomial equations over Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For each graph Gs in our  6  family,  �  ij∈E(Gs)  xixj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In addition, add the linear equation �n  i=1 xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  The sum of the degrees of the polynomials here is 2(n/2 − 1) + 1 = n − 1, which  is smaller than the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Since the system is homogeneous it admits  the trivial solution xi = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Any other solution (which exists by the Chevalley  Warning Theorem) gives a clique on the set of vertices {i : xi = 1} which is nonempty, of  even cardinality, and contains an even number of edges (possibly zero) of each Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This  establishes the lower bound and completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  4  Concluding remarks and open problems  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1, which is equivalent to the problem of deciding whether or not for any  fixed nonempty graph H with an even number of edges dH(n) tends to 0 as n tends  to infinity, remains wide open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  An interesting special case is whether or not dK4(n) = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It is also interesting  to decide whether or not dK4(n) ≥  1  no(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It is not difficult to show that the latter  would follow from the existence (if true) of an edge coloring of Kn by no(1) colors  with no copy of K4 in which every color appears an even number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This may  be related to the construction in [10], see also [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Gowers conjectured in [8] that any family of a constant fraction of all graphs on [n],  where n is sufficiently large, contains two graphs G1, G2 such that G2 is a subgraph  of G1 and the symmetric difference of the two graphs (that is, the set of all edges of  G1 that are not in G2) forms a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This is clearly stronger than the conjecture  that dK(n) tends to 0 as n tends to infinity, which is also open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' As explained in  [8] the question of Gowers can be viewed as the first unknown case of a polynomial  version of the density Hales-Jewett Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  As mentioned in the remark following the statement of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='5, it is not  difficult to show that for every graph H with an even number of eges the maximum  possible cardinality of a linear family of graphs on [n] in which no symmetric differ-  ence is a copy of H, is o(2(n  2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' As the proof applies Ramsey’s Theorem, it provides  very weak bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It will be interesting to establish tighter bounds for the linear  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6 provides an example of a tight result of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  The problem considered above can be extended to hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' More generally, it  can be extended to other versions of problems about binary codes, where the coordi-  nates of each codeword are indexed by the elements of some combinatorial structure,  7  and the forbidden symmetric differences correspond to a prescribed family of sub-  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Here is an example of a problem of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' What is the maximum  possible cardinality of a collection of binary vectors whose coordinates are indexed  by the elements of the ordered set [n], where no symmetric difference of two dis-  tinct members of the collection forms an interval of length which is a cube of an  integer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The corresponding Cayley graph here has 2n vertices, and it is triangle-free  by Fermat’s last Theorem for cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Its independece number, which is the answer  to the question above, is o(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Indeed, this follows from the Furstenberg-S´ark¨ozy  Theorem and its extensions [11], by considering the maximum possible cardinality  of an independent set in the induced subgraph on the set of all vertices that are  characteristic vectors of an interval [i] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
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OdAyT4oBgHgl3EQf7PpY/content/tmp_files/2301.00835v1.pdf.txt

@@ -0,0 +1,3589 @@
+Timed Model-Based Mutation Operators for Simulink Models
+Jian Chen* 1 Manar H. Alalfi 2 Thomas R. Dean 3
+1 Department of Electrical and Computer Engineering, Queen’s University
+Kingston, ON, Canada
+E-mail: [email protected]
+2 Department of Computer Science, Ryerson University
+Toronto, ON, Canada
+E-mail: manar.alalfi@cs.ryerson.ca
+3 Department of Electrical and Computer Engineering, Queen’s University
+Kingston, ON, Canada
+E-mail: [email protected]
+Abstract
+Model-based mutation analysis is a recent research area, and real-time system testing can benefit from
+using model mutants. Model-based mutation testing (MBMT) is a particular branch of model-based test-
+ing. It generates faulty versions of a model using mutation operators to evaluate and improve test cases.
+Mutation testing is an effective way to ensure software correctness and has been applied to various appli-
+cation areas. Simulink is a vital modeling language for real-time systems. This paper introduces Simulink
+model mutation analysis to improve Model-in-the-loop (MIL) testing. We propose a set of Simulink mu-
+tation operators based on AUTOSAR, which reflects the temporal correctness when a Simulink model
+is mapped to Operating System tasks. We implement a mutation framework that generates mutants for
+implicit clock Simulink models. Finally, we demonstrate how this framework generates mutants to reveal
+task interference issues in the simulation. Our work integrates the Simulink model with the timed systems
+to better support mutation testing automation.
+Keywords: Mutation Testing, Model-Based Testing, Model-Based Mutation Testing, Mutation Operator,
+Simulink, Real-Time System, Scheduling, AUTOSAR
+1.
+Introduction
+Today, cars come equipped with advanced technolo-
+gies that did not exist before, such as Automatic
+Emergency Braking (AEB), Adaptive Cruise Con-
+trol (ACC), Lane Departure Warning/Lane Keeping,
+and Autonomous driving. All of these features rely
+on software to realize sophisticated control algo-
+rithms. Generally, such software is developed within
+the timed system context, in which the system cor-
+rectness not only relies on the software implemented
+functions correctness but also depends on the sys-
+tem to meet time constraints. Many factors can con-
+tribute to the execution time of a system running on
+a target platform. Issues such as task interference
+may cause delays during task execution. Software
+quality plays a crucial role in such safety-critical ap-
+plications.
+Model-Based Testing (MBT) is a promising
+technique for the automated testing of timed sys-
+arXiv:2301.00835v1  [cs.SE]  2 Jan 2023
+
+J. Chen et al. / Mutation Operators for Simulink Models
+tems. A model represents the behavior of software,
+and the model is usually abstracted from real-time
+specifications. However, some modeling environ-
+ments support this feature in the Hardware-in-the-
+loop (HIL) simulation testing instead of the MIL.
+For example, Matlab/Simulink (ML/SL) simulations
+assume block behaviors are completed in nearly zero
+execution time, while real execution requires a finite
+execution time, which may cause a failure. ML/SL
+models are based on the Synchronous Reactive (SR)
+model 23 that may assume the task execution times
+are zero. Errors in the model may not be apparent
+without an explicit real-time execution in the MIL
+phase. Usually, a Simulink model can be well simu-
+lated in the MIL, but it may have errors in the real-
+time context.
+Hence, MBT needs an extension to accommo-
+date the real-time context, which includes modeling
+the system through a timed formalism, and check-
+ing the implementation conforms to its specification.
+Traditionally, this is done via conformance checks
+35. Recently, several tools have been proposed to
+simulate the real-time execution effects for ML/SL
+models in MIL, such as TrueTime 21, TRES 12,
+Timing-aware blocks 27, and SimSched 9. SimSched
+uses a model transformation to integrate scheduling
+into the model to validate the real-time context dur-
+ing simulation. To evaluate SimSched, we turn to
+mutation testing using mutation analysis to assist the
+evaluation of the SimSched tool.
+In this paper, we propose a set of mutation op-
+erators with a timed task model, which is based
+on the AUTomotive Open System ARchitecture
+(AUTOSAR), that reflects the temporal correctness
+when a Simulink model is mapped to Real-Time Op-
+erating System (RTOS) tasks in a real-time context.
+This paper is organized as follows:
+Section
+2 introduces background information.
+Section 3
+presents the set of proposed timed mutation oper-
+ators for Simulink models. Section 4 explains the
+usage of the timed mutation operators. Section 5
+presents validation experiments and results. Section
+6 summarizes related studies in MBT. Finally, Sec-
+tion 7 presents the conclusions of our work and out-
+lines future work.
+2.
+Background
+This section gives an overview of the background
+information on the material needed to explain our
+work. We begin with a basic introduction to mu-
+tation testing, Simulink, and AUTOSAR; then, we
+present our timed task model.
+2.1.
+Mutation testing
+Mutation testing was introduced in the 1970s 17,13,22
+and proved to be an effective way to reveal software
+faults 32. It is a fault-based software testing tech-
+nique, which has been extensively studied and used
+for decades. It contributes a range of methods, tools,
+and reliable results for software testing. Mutation
+testing is designed to find valid test cases and dis-
+cover real errors in the program.
+Model-Based Mutation Testing (MBMT) takes
+the advantages of both model-based testing and mu-
+tation testing and has been widely applied to mul-
+tiple types of models such as feature models 20,
+statechart-based models 41,1, timed automata 3,2, and
+Simulink 8,26,38. However, in real-time system de-
+velopment, both logical and temporal correctness is
+crucial to the correct system functionality. The tem-
+poral correctness depends on timing assumptions for
+each task. Timed Automata (TA) 4 is a common for-
+malism to model and verify real-time systems to see
+whether designs meet temporal requirements. Aich-
+ernig et al.3 propose an MBMT technique for timed
+automata that applies to input/output timed automata
+(TAIO) model. Nilsson et al.
+28 add an extension
+to the TA formalism with a task model, and their
+mutation operators focus on timeliness. Simulink∗is
+widely used for model-driven development of soft-
+ware within the automotive sector. Most of the mu-
+tation operators proposed for Simulink models are
+from a property point of view either run-time or
+design-time such as signal modification, arithmetic
+alternation, or block change 18,38,36,43. Some of the
+proposed mutation testings are targeted at test case
+generation for Simulink models 8,19. However, there
+is no mutation operator with an explicit clock model
+for Simulink.
+* https://www.mathworks.com/products/simulink.html
+
+J. Chen et al. / Mutation Operators for Simulink Models
+2.2.
+Simulink
+Simulink is one of the most popular modeling lan-
+guages for modeling dynamical systems, and MAT-
+LAB provides a graphical programming environ-
+ment to perform system simulations. Simulink mod-
+els are graphical blocks and lines, and they are con-
+nected by signals between input and output ports.
+The Simulink simulation engine determines the ex-
+ecution order of blocks based on the data depen-
+dencies among the blocks before a simulation exe-
+cution. Simulink defines two types of blocks, di-
+rect feedthrough, and non-direct feedthrough, to as-
+sure the correct data dependencies in the simula-
+tion. Simulink uses the following two basic rules
+25 to determine the sorted execution order: A block
+must be executed before any of the blocks whose
+direct-feedthrough ports it drives; Blocks without
+direct feedthrough inputs can execute in arbitrary or-
+der as long as they precede any block whose direct-
+feedthrough inputs they drive. All blocks are sched-
+uled in sorted order and executed in sequential exe-
+cution order. The Simulink engine maintains a vir-
+tual clock to execute each ordered block at each vir-
+tual time.
+Simulink Coder†supports code generation and of-
+fers a framework to execute the generated code in
+a real-time environment. Simulink Coder can gen-
+erate code for the periodic task, either using a sin-
+gle task or a multi-task. Single-task implementa-
+tions can preserve the semantics during the simula-
+tion because the generated code is invoked by a sim-
+ple scheduler in a single thread without preemptions.
+For multi-task implementations, the generated code
+is invoked by a rate monotonic (RM) 24 scheduler
+in a multithreaded RTOS environment, where each
+task is assigned a priority and preemptions occur be-
+tween tasks. As a consequence of preemption and
+scheduling, the implementation semantic can con-
+flict with the model semantic in a multi-rate system.
+2.3.
+AUTOSAR
+AUTOSAR is an open industry standard to meet the
+needs of future car development. AUTOSAR de-
+fines three main layers: the application, the runtime
+environment (RTE), and the basic software (BSW)
+layer 40. The functions in the application layer are
+implemented by SW-Cs, which encapsulate part or
+all of the automotive electronic functions, as shown
+in Figure 1. The components communicate via a
+Virtual Functional Bus (VFB), which is an abstrac-
+tion of all the communication mechanisms of AU-
+TOSAR. Engineers abstract the communication de-
+tails of software components employing VFBs. A
+set of runnables represents the SW-Cs internal be-
+haviors, and a runnable is the smallest executable
+code that can be individually scheduled, either by
+a timer or an event. Lastly, runnables are required
+to map to a set of tasks for a target platform, and
+the mapping has to preserve ordering relations and
+causal dependencies. Simulink has supported AU-
+TOSAR compliant code generation since version
+R2006a‡.
+All AUTOSAR concepts can be repre-
+sented by Simulink blocks and the existing Simulink
+blocks can be easily used in the AUTOSAR devel-
+opment process. Some of AUTOSAR concepts and
+Simulink concepts mapping relation is shown in Ta-
+ble 1 39.
+Fig. 1. AUTOSAR components, interfaces, and runnables.
+(Adapted from 5)
+† https://www.mathworks.com/products/simulink-coder.html
+‡ https://www.mathworks.com/products/simulink.html
+
+SW-C 1
+SW-C 2
+SW-C 3
+SW-C n
+Runnable 2a
+Runnable 3a
+Runnable na
+Runnable 1a 
+！
+Runnable 2b
+Runnable nb
+NTOIA
+V
+OIDID
+Virtual Function Bus(VFB)
+Tool supporting deployment
+System
+ECU
+of sW components
+Constraints
+Descriptions
+个
+ECU I
+ECU II
+ECU M
+SW-C 1
+SW-C 3
+SW-C 2
+SW-C n
+RTE
+RTE
+RTE
+Basic Software
+Basic Software
+Basic SoftwareJ. Chen et al. / Mutation Operators for Simulink Models
+Table 1. Examples of ML/SL and AUTOSAR Concepts Map-
+ping.
+ML/SL
+AUTOSAR
+Subsystem
+Atomic Software
+Component
+Function-call subsystem
+Runnable
+Function calls
+RTEEvents
+2.4.
+Task model
+In automotive software, Simulink models are often
+drawn from real-time specifications and are realized
+as a set of tasks running on an RTOS. In order to
+better test this kind of software in the MIL phase,
+model-based testing needs to be scaled to the real-
+time context, which includes a timed formalism to
+model the system under test conforming with the
+real-time requirements. We define a task model to
+model the timing properties of tasks in the Simulink
+environment and the application is modeled as a set
+of periodic tasks.
+A task model, T, is represented by a tuple
+{φ,ρ,c,γ, prect, precr, prio, jitter}, where φ is an
+offset of the task, ρ is the period of the task, c is the
+Worst Case Execution Time (WCET) of the task, γ
+is a list of runnables that belong to the task, prect is
+the precedence constraint of the task, precr is the
+precedence constraint of the runnables within the
+task, prio is the priority associated with the task,
+and jitter is the deviation of a task from the periodic
+release times. Every task has an implicit deadline
+which means the deadline of a task is equal to ρ. An
+offset φ refers to the time delay between the arrival
+of the first instance of a periodic task and its release
+time. A WCET is the summation of each runnable
+execution time. A precedence constraint prect is a
+list of tasks that specifies the task execution order,
+and precr is a list of runnables within a task.
+Fig. 2. Task states and transitions of task model.
+Figure 2 shows the task-state and transition dia-
+grams of the task model that is based on OSEK’s ba-
+sic task-state model. The task model includes three
+states: suspended, ready, and running, and four tran-
+sitions: Active, Stare, Preempt, and Terminate. The
+transitions represent the actions to activate, start,
+preempt, or terminate a task.
+Fig. 3. Task timing parameters shown in Gantt chart (all
+related to Task2).
+Figure 3 shows the timing parameters of a task
+model and different timing parameters can alter the
+application’s real-time behavior within a system.
+3.
+Mutation Operators for Simulink Model
+This section introduces a mutation analysis ap-
+proach to validate real-time context during the sim-
+ulation. Mutation operators are the key elements of
+mutation testing. Model-based mutation testing is
+a method of injecting faults into models to check
+whether the tests are passed or failed, thus validat-
+ing the software. The injecting faults are the muta-
+tion operators. Before we apply mutation operators
+to the model, we need to identify them which is to
+
+Start
+Ready
+Running
+Preempt
+Active
+Terminate
+SuspendedPriority
+Active
+Start
+Preempt
+Terminate
+Task1
+Task2
+offseti
+C1
+C2
+jitter
+period
+TimeJ. Chen et al. / Mutation Operators for Simulink Models
+understand what kind of errors can cause failure. We
+have proposed the following task-related mutation
+operators.
+3.1.
+Offset mutation operators
+The task release offset is one of the factors that affect
+the computation result in terms of task interference.
+In order to take the offset into account for analy-
+sis, we introduced an offset mutation operator. For a
+known offset φ, a task can now execute after φ time
+units with respect to the start of its period. The exe-
+cution time of the task is unchanged at c time units
+before the next period starts.
+3.1.1. mITO
+This operator adds δ time to the current offset. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the offset φi to
+φi +δ.
+3.1.2. mDTO
+This operator subtracts δ time to the current offset.
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the offset φi to
+φi −δ.
+3.2.
+Period mutation operators
+An RTOS usually applies a preemptive multitask-
+ing scheduling algorithm to determine the execu-
+tion order of tasks, and the most picked algorithm is
+fixed-priority scheduling (FPS). The algorithm as-
+signs each task a static priority level. The RTOS
+scheduler executes the highest priority task from the
+ready task queue. Simulink Coder supports an RM
+scheduler, where the priority of a task is associated
+with its period, if a task has a smaller period, then it
+has a higher priority. Furthermore, a lower-priority
+task can be preempted by a more top-priority task
+during the execution.
+3.2.1. mITPER
+This operator increases the period of a task, which
+changes the task to a slower rate. For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the period of the task i
+to ρi +δ.
+3.2.2. mDTPER
+This operator decreases the period of a task, which
+changes the task to a faster rate. For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the period of the task i
+to ρi −δ.
+3.3.
+Execution time mutation operators
+The correctness of a real-time system is determined
+on one hand by the computation results of the log-
+ical program, and on the other hand, is strictly re-
+lated to the time at which the results are produced.
+Hence, execution time analysis is essential during
+the process of designing and verifying embedded
+systems. For this reason, we propose execution time
+operators, which can adjust the execution time of
+each task at the runnable level to simulate the run
+time execution on different processor speeds. The
+longer execution time of a task may lead to a sce-
+nario where a lower-rate task blocks a higher-rate
+task so that it misses its deadline.
+3.3.1. mITET
+This operator adds δ time to the current exe-
+cution time of each runnable, which increases
+the total execution time.
+For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the execution time ci to
+ci +δ.
+3.3.2.
+mDTET
+This operator subtracts δ
+time from the cur-
+rent execution time of each runnable, which de-
+creases the total execution time. For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the execution time ci to
+ci −δ.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+3.4.
+Execution precedence mutation operators
+The RTOS scheduler selects tasks to execute accord-
+ing to the priority level of the task. However, the
+spawn order determines the execution order of tasks
+with the same priority. Whichever task is spawned
+first is realized and gets the CPU first to run. This re-
+sults in situations where a pair of tasks have a prece-
+dence relation in the implementation that does not
+exist in the design phase lost an existing precedence
+relation in the implementation. The incorrect prece-
+dence can cause a wrong execution order of tasks.
+Hence, we proposed the precedence mutation oper-
+ators which can specify a precedence relation be-
+tween a pair of tasks and runnables. This operator
+creates mutants by assigning a specific execution or-
+der to a set of tasks or runnable to reflect the prece-
+dence relation.
+3.4.1. mATPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each task τ j ∈ T (j ̸= i), if τj /∈
+precti, this mutation operator adds τj to precti.
+3.4.2. mRTPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each task τj ∈ T (j ̸= i), if
+τj ∈ precti, this mutation operator removes τ j from
+precti.
+3.4.3. mARPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each runnable γim ∈ τi, if γim /∈
+precri, this mutation operator adds γim to precri.
+3.4.4. mRRPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each runnable γim ∈ τi, if γim ∈
+precri, this mutation operator removes γim from
+precri.
+3.5.
+Priority mutation operators
+In an RTOS, each task is assigned a relative priority,
+which is a static integer to identify the degree of im-
+portance of tasks. The highest priority task always
+gets the CPU when it becomes ready to run. The
+most common RTOS scheduling algorithm is pre-
+emptive scheduling.
+3.5.1. mITPRI
+This operator increases the priority of a task. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the priority of the
+task prioi to proii +δ.
+3.5.2. mDTPRI
+This operator decreases the priority of a task. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the period of the
+task i to proii −δ.
+3.6.
+Jitter mutation operators
+Timing jitter exists in the RTOS, and it is the delay
+between subsequent periods of time for a given task.
+3.6.1. mITJ
+This operator increases the jitter time of a task. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the priority of the
+task jitteri to jitteri +δ.
+3.6.2.
+mDTJ
+This operator decreases the jitter time of a task. For
+a given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the period of the
+task jitteri to jitteri −δ.
+3.7.
+Shared memory mutation operators
+It is common that RTOS tasks exchange data or
+information via shared memory(e.g., global vari-
+able, memory buffer, hardware register). The shared
+memory can easily cause access conflict if the logi-
+cal software design is neglected in any corner case.
+Here we introduce a set of variable mutation opera-
+tors.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+3.7.1. mDSM
+This operator defines a new value to a global vari-
+able in a task. If a task reads this global variable,
+then we define a new value right before the reads
+occurred.
+3.7.2. mUDSM
+This operator un-defines a global variable in a task.
+If a task writes this global variable, then ignore this
+writes operation.
+3.7.3. mRDSM
+This operator removes the definition of a global vari-
+able. If a global variable is initialized in a task then
+do not initialize it.
+3.7.4. mRSM
+This operator adds a reference to a global variable.
+3.7.5. mRMSMR
+This operator removes reference to a global variable.
+3.7.6. mRSMR
+This operator replaces a reference of a global vari-
+able with a different global variable.
+Table 2. Simuilnk Mutation Operators
+Mutation Key
+Title
+mITO
+Increase Task Offset
+mDTO
+Decrease Task Offset
+mITPER
+Increase Task Period
+mDTPER
+Decrease Task Period
+mITET
+Increase Task Execution Time
+mDTET
+Decrease Task Execution Time
+mATPREC
+Add Task Precedence
+mRTPREC
+Remove Task Precedence
+mARPREC
+Add Runnable Precedence
+mRRPREC
+Remove Runnable Precedence
+mITPRI
+Increase Task Priority
+mDTPRI
+Decrease Task Priority
+mITJ
+Increase Task Jitter
+mDTJ
+Decrease Task Jitter
+mDSM
+Define Shared Memory
+mUDSM
+Un-define Shared Memory
+mRDSM
+Remove Definition Shared Memory
+mRSM
+Reference a Shared Memory
+mRMSMR
+Remove a Shared Memory Reference
+mRSMR
+Replace a Shared Memory Reference
+4.
+Mutation operators demonstration
+We have introduced twenty mutation operators cate-
+gorized into seven classes and explained each mu-
+tation class.
+The mutation operators are summa-
+rized in Table 2. We use simple examples to demon-
+strate the use of each mutation operator. To demon-
+strate our mutation operators, we use the tool Sim-
+Sched to alter the properties of software applications
+realized as Simulink models.
+From Table 1, we
+know that each function-call subsystem represents
+an AUTOSAR runnable. The function-call subsys-
+tem can be executed conditionally when a function-
+call event signal arrives. Both an S-function block
+and a Stateflow block can provide such a function-
+call event. SimSched applies the function-call in-
+vocation mechanism to use an S-function to gener-
+ate a function-call event to schedule each runnable
+(function-call subsystem). Figure 4 shows the Sim-
+Sched parameters dialogue that we can utilize it to
+adjust the timing properties to implement the muta-
+tion operator for Simulink models.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 4. SimSched Parameter setting dialogue.
+In this section, we use several simple examples
+to exhibit the mutants generated by our mutation op-
+erators. Figure 5 illustrates the use of SimSched to
+schedule a Simulink model. In this example, Sim-
+Sched is at the top left corner, which schedules three
+runnables (R1, R2, R3), and they are mapped to two
+tasks (T1, T2). Runnable R1 is mapped to T1, the pe-
+riod of T1 is 10ms, priority is 2, and the execution
+time of R1 is 3ms. R2 and R3 are mapped to T2. The
+period of T2 is 20ms, priority is 1, and the execu-
+tion time of R2 and R3 are 3ms and 3ms accordingly.
+The detailed parameter settings are listed in Table 3.
+There is a Data Store Memory block in this exam-
+ple, named A, which defines a shared data store that
+is a memory area used by Data Store Read and Data
+Store Write block with the same data store name. R1
+writes a constant value to a global variable A. R2
+reads A first then writes the summation of A and its
+delay value to A. R3 reads A then subtracts its delay
+value from A, and outputs the result.
+Table 3. The simple example settings
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+10
+3
+2
+R1
+T2
+20
+3
+1
+R2
+T2
+20
+3
+1
+R3
+Fig. 5. A simple example of using SimSched to schedule
+AUTOSAR SW-Cs.
+Fig. 6. Task executions Gantt chart of the running example.
+Fig. 7. A simple example of using Stateflow to schedule
+AUTOSAR SW-Cs.
+Fig. 8. Simple example output of Stateflow scheduler
+simulation.
+
+BlockParameters:SimSched
+S-Function (mask)
+Parameters
+Task:
+[1, 2]
+Priority:
+[2,1]
+Period:
+[10,20]
+Runnable:
+"R1','R2',"'R3'
+[1, 2, 2]
+Task Mapping:
+Execution Time:
+[3, 3, 3]
+[0,0,0]
+Offset
+[0,0,0]
+Jitter
+OK
+Cancel
+Help
+ApplyScheduler
+Runnable(period, execution time, priority) Task()
+R1(10ms, 3ms, 2)
+ Task(1)
+R2( 20ms, 3ms,
+ Task(2)
+R3 20ms, 3ms,
+Task(2)
+function()
+function()
+function(
+SimSched
+function
+Runnable1 subsystem
+Runnable2 subsystem
+Runnable3 subsystem1
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+1
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.060:0
+call()
+0:1
+R1()
+1 ms Clock
+R2()
+0:1
+R3()
+call(
+Temporal Logic
+0:6
+Scheduler
+0:1
+0:1
+0:1
+R1()
+R2()
+R3()
+R1
+0:7
+A
+subrater
+Runnable1 subsystem1
+Runnable2 subsystem1
+Runnable3 subsystem'11
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+40
+R2
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+20
+10
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+TimeJ. Chen et al. / Mutation Operators for Simulink Models
+We use a Stateflow scheduler version of the sim-
+ple example shown in Figure 7 to show the typical
+Stateflow scheduler simulation result, then compare
+it with the SimSched scheduler simulation result.
+The task parameters are all the same shown in Table
+3. We apply the same task configurations for both
+the Stateflow scheduler and SimSched models for
+simulation. Figure 8 shows the Stateflow scheduler
+simulation result, and Figure 9 shows the SimSched
+simulation result. The result figures show the output
+value of each runnable. From the figure, we can see
+that R1, R2 and R3 are all executed at time 0 in Fig-
+ure 8; R1, is executed time 0, R2 is executed at 3ms,
+and R3 is executed at 6ms in Figure 9. R2 and R3 are
+executed later than the Stateflow scheduler simula-
+tion in Figure 8. This is because that SimSched takes
+into account execution time, and each task must be
+executed until the previous task is completed on a
+single core platform.
+Fig. 9. Simple example output of SimSched simulation without
+applying any mutation operator.
+4.1.
+Offset mutation operators
+We first apply the mITO mutation operator to the
+running example, let’s say that increase δ1 = 3ms
+for T1 then we have the task execution timeline in
+Figure 10. We can see T2 is executed first at time
+0, and T1 preempts T2 at 3ms in the first period due
+to the offset effect. After the first period, there is
+no preemption between T1 and T2. Then, we apply
+the mDTO mutation operator based on the previous
+settings. We set δ1 = −1ms to T1 then the offset
+for T1 is 2ms. Figure 11 shows the task execution
+timeline. T2 is preempted by T1 during the execution
+of the first period. Compared to the task execution
+Gantt chart of our running example shown in Figure
+6 with no offset, we can clearly see the preemption
+effect.
+Fig. 10. Task executions Gantt chart of the running example
+after increase offset mutation operator is applied.
+Fig. 11. Task executions Gantt chart of the running example
+after decrease offset mutation operator is applied.
+The running example’s output after applying the
+mITO mutation operator is shown in Figure 12
+which is different from Figure 9. Because T1 pre-
+empts T2 at the first instance execution, the output
+of R3 is from zero to ten then goes back to zero
+then goes up instead of always increasing value. The
+running example’s output after applying the mDTO
+mutation operator is the same as Figure 9 because
+the offset operator only affects the initial execution
+of each task, and the preemption occurs before the
+first execution of R2 instance completion. Inside our
+model scheduler program, we trigger each subsys-
+tem at the end of each execution time slot. Techni-
+cally, the execution order of this example is still R1,
+R2, and R3 so the output of the simulation keeps the
+same.
+Fig. 12. Simple example output of SimSched simulation after
+applying mITO mutation operator.
+
+T1
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05
+0.06
+Time(sec.)
+T2
+0.5
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)T1
+06
+0
+0
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+T2
+0.5
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)Runnable1 subsystem
+11
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+Runnable2_subsystem
+20
+10
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+Runnable3_subsystem
+20
+10
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)11
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+30
+20
+oR2 
+0
+0.003
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+20
+10
+0
+0
+0.006
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+TimeJ. Chen et al. / Mutation Operators for Simulink Models
+4.2.
+Period mutation operators
+We apply the mITPER operator to this example to
+increase the period of a task. We set δ1 = 1ms to T1
+so the period of the task T1 is 11ms now. Figure 13
+shows that T2 is preempted at the time of 22ms, and
+the simulation yields a wrong result due to this pre-
+emption shown in Figure 14. The output of R3 is an
+alternating value instead of an increasing value.
+Fig. 13. Task executions Gantt chart of the running example
+after increase period mutation operator is applied.
+Fig. 14. Simple example output of SimSched simulation after
+applying mITPER mutation operator.
+We apply the mDTPER operator to this example
+to decrease the period of a task. We set δ1 = −4ms
+to T1 so the period of the task T1 is 6ms now. Then,
+we run the simulation, T2 is preempted by T1 shown
+in Figure 15 and it yields a wrong simulation result
+shown in Figure 16. The output of R3 is either zero
+or ten instead of an increasing value.
+Fig. 15. Task executions Gantt chart of the running example
+after decreasing period mutation operator is applied.
+Fig. 16. Simple example output of SimSched simulation
+after applying mDTPER mutation operator.
+4.3.
+Execution time mutation operators
+We apply the mITET operator to this example to in-
+crease the execution time of a task. We can specify
+any runnable to increase its execution time within a
+task. For example, we set δ1 = 4ms to R2 in T2 so
+the execution of R2 is 7ms and T2 takes 10ms to ex-
+ecute now. Figure 17 shows that T2 is preempted at
+the time of 10ms, and the simulation yields a wrong
+result due to this preemption. The wrong result is
+the same as the example of applying decreasing task
+period. We apply the mDTET operator to this exam-
+ple to decrease the execution time of a task. We set
+δ1 = −1ms to T1 so the execution time of the task T1
+is 2ms now. Then, we run the simulation, there is no
+preemption that occurs between these two tasks and
+the output is as expected as the original model.
+Fig. 17. Task executions Gantt chart of the running example
+after increase execution time mutation operator is applied.
+4.4.
+Execution precedence mutation operators
+We introduce the second example as Table 4 to ex-
+plain the mATPREC and mRTPREC operators. Fig-
+ure 18 shows the task execution Gantt chart of this
+example. From the task execution chart, we can see
+the execution order of the tasks is T1,T2,T1,T3.
+
+1
+0.5
+0E
+0
+0.01
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)Runnable1_sybsystem
+11
+10
+9 
+0.1
+0.2
+0.3
+0.4
+0.5 
+0.6
+0.7 
+0
+0.8
+0.9
+Runnable2_subsystem
+400
+200
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Runnable3_subsystem
+400 
+200
+0
+0
+0.1
+0.2
+0.3
+9:0
+0.0
+0.8
+0.9T1
+1
+0.5 
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+T2
+1
+0.5 
+0
+0
+0.02
+0.03
+0.04
+0.05
+0.06Runnable1_sybsystem
+11
+10
+9.
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+Runnable2_subsystem
+40 
+20 
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+Runnable3_subsystem
+10 
+5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.11
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05 
+0.06
+1
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06J. Chen et al. / Mutation Operators for Simulink Models
+Table 4. The simple example settings
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+5
+1
+3
+R1
+T2
+10
+4
+2
+R2
+T3
+10
+3
+1
+R3
+Fig. 18. Task executions Gantt chart of example 2.
+First, we assume there is no precedence rela-
+tion among tasks so we use the mATPREC mutation
+operator to add a precedence relation τ3 to prect2,
+which specifies that a new instance of T2 cannot start
+unless T3 has executed after the last instance of T2.
+Hence, we set the execution order that T3 is executed
+before T2 in the setting dialogue. Figure 19 shows
+the execution result that T2 is preempted by T1. If T2
+is not a re-entrant function then this preemption may
+cause potential failure execution.
+Fig. 19. Task executions Gantt chart of example 2 after task
+precedence mutation operator is applied.
+Then, we assume there is a precedence relation
+between T1 and T3 and the task execution diagram
+is the same in Figure 18. We apply the mRTPREC
+mutation operator to remove the precedence relation
+prect3 from τ1. The result is the same as shown in
+Figure 19.
+We add one runnable R4 to the first example and
+assign it to T1. This new task configuration is shown
+in Table 5. R4 writes a different constant value from
+R1 to the global variable A. We apply mARPREC
+mutation operator to this new example, which adds
+γ1 to precr4. R4 requires R1 execute first so R4 over-
+writes the value written by R1. The operator changes
+the execution order of runnables.
+Table 5. Task configuration settings for runnable precedence
+mutation operators.
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+10
+2
+2
+R1
+T2
+20
+2
+1
+R2
+T3
+20
+2
+1
+R3
+T1
+10
+2
+1
+R4
+In example one, T2 has two runnables R2 and R3
+with a precedence relation between them. We apply
+mRRPREC runnable remove precedence mutation
+operator to remove the precedence γ2 from precr3.
+We schedule R3 runs before R2 since no precedence
+constraint that turns out different than the original
+simulation. The original output of R3 is an increas-
+ing value along with the execution instead of a value
+of either zero or a fixed value. The reason is that
+R3 executes first and it reads A before R2 writes any
+new value to A. The bottom output line in Figure 20
+shows the execution result.
+Fig. 20. The outputs of example one three runnables.
+4.5.
+Priority mutation operators
+Table 6. Priority mutation operator example settings
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+10
+1
+4
+R1
+T2
+10
+2
+3
+R2
+T3
+10
+3
+2
+R3
+We apply mITPRI operator to the example in Ta-
+ble 6 to increase the priority of T3. This mutation
+operator changes the priority of prio3 to proi3 + 3
+so the T3 has the highest priority 5 in this example,
+which results in T3 being executed at first. Figure 21
+shows T3 is triggered first in the task execution Gantt
+chart. This mutation alters the task execution order.
+
+1
+口
+口
+口
+口
+口
+口
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05 
+0.06
+1
+0
+0
+0. b1
+0.02 
+0.03
+0.04 
+0.05
+0.06
+1
+1
+1
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.061 
+0.5
+0
+4.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+0.5
+0
+0
+0. 01
+0.02
+0.03
+0.04 
+0.05
+0.06
+L
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.0611
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+40
+20 
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+10 F
+5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 21. Task executions Gantt chart after applying increas-
+ing task priority mutation operator.
+We apply mDTPRI operator to decrease the pri-
+ority of T1. This mutation operator changes the pri-
+ority of prioi to proi−3 so the T1 has the lowest pri-
+ority 1 in this example, which results in T1 being
+executed at last. The task execution Gantt chart is
+shown in Figure 22.
+Fig. 22. Task executions Gantt chart after applying decreas-
+ing task priority mutation operator.
+4.6.
+Jitter mutation operators
+We apply mITJ operator to increase a jitter time of
+a task. For example, let δ = 2, this mutation op-
+erator changes the real release time of the task to
+jitter1 = 0+2. Figure 23 shows the execution of T2
+is preempted by T1 caused by the jitter.
+Fig. 23. Task executions Gantt chart after applying increas-
+ing jitter mutation operator.
+Then mDTJ mutation operator decreases the jit-
+ter time of a task. We apply this operator to the
+above example and let δ = −1 so the task jitter1 =
+2 − 1. T2 is preempted by T1 during the simulation
+phase.
+4.7.
+Shared memory mutation operators
+In this shared memory category, we introduce five
+mutation operators.
+The first one is mDSM, and
+this operator assigns a new value to the memory
+store before a read.
+For our example, we add a
+Data Store Write block right before the Data Store
+Read execution so that the Data Store Write block
+defines a new value to the variable, and we chose
+the initial value of this variable as the default new
+value. The mutant using mUDSM operator is shown
+in Figure 24, which only shows the changes of
+Runnable2 subsystem. We add a constant block and
+a Data Store Write block at the top left corner.
+Fig. 24. A simple example of DSM mutant.
+The second mutant operator is mUDSM, and this
+operator disregards a write to a Data Store block.
+For our example, we remove the Data Store Write
+block. Figure 25 shows the mUDSM mutant that the
+Data Store Write has been removed.
+Fig. 25. A simple example of UDSM mutant.
+The third mutant operator is mRDSM, and this
+operator removes an initialization value to a Data
+Store memory. In many programs, variables require
+
+1F
+1
+0.5 
+0
+/o
+0.01
+ 0.02 
+0.03
+ 0.04 
+0.05 
+0.06
+1
+1
+0.5 
+0
+0.01
+0.02 
+0.03 
+ 0.04 
+0.05
+0.06 
+0.5
+0.01
+0.02 
+0.03 
+0.04 
+0.05
+0.061 F
+0.5
+0
+/
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+1 
+0.01
+0
+0.02
+0.03
+0.04 
+0.05
+0.06
+0.5
+0
+0.01
+0.02
+0.03
+0.04 
+0.05
+0.061
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05
+0.06
+0.5
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06f()
+2:0
+2:1
+function
+A
+0
+2:4
+2:2
+A
+2:5
+A
+2:3
+Zf()
+function
+A
+1
+Z.J. Chen et al. / Mutation Operators for Simulink Models
+an initial value before they can use properly. For
+our example, Runnable1 subsystem is such a pro-
+cess of initializing Data Store A; then, we remove
+the Data Store Write block in Runnable1 subsystem.
+The Simulink model can still run simulations with-
+out any issues however the output of the simulation
+only yields a single value.
+Mutant operator mRSM adds a new reference to
+shared memory. Figure 26 shows the block diagrams
+of a Simulink model with three subsystems and they
+are mapped to two tasks. The model has a DSM
+block A in the root-level system. There is a Data
+Store Write block inside subsystems Task B1 and a
+Data Store Read block in Task B2. The period of
+Task A is 5ms and the period of Task B is 10ms.
+To implement the mRSM, we add a Data Store Read
+block to the TaskA subsystem which shows in Figure
+27. In the original example, Task A executes first
+then Task B1 writes A and Task B2 reads A. The
+mutant program has the same execution order as the
+original model. However, when the Data Store Read
+block in Task A executes, the block reads data from
+an uninitialized data store or a previous instant of
+Task B1 as Task B has not executed yet or has been
+executed previously.
+Fig. 26. A simple Simulink model.
+Fig. 27. An example of mRSM mutant operator. Adding a
+Data Store Read block to Task A block.
+Mutant operator mRMSMR deletes a reference to
+shared memory. In Figure 26, Task B2 has a refer-
+ence to a DSM block A in the root-level system. To
+implement the mRMSMR, we delete the Data Store
+Read block in the TaskB2 subsystem. In the mu-
+tant program, Task B2 has a constant output value
+of zero since there is no reference.
+5.
+Evaluation Phase
+In the previous section, we describe how a model
+scheduler SimSched can validate the real-time con-
+text during a simulation, and we utilize mutation
+testing to evaluate SimSched. In this section, we
+perform experiments to demonstrate the use of our
+mutation testing framework to evaluate the quality
+of SimSched and Stateflow schedulers in scheduling
+tasks in real-time systems.
+5.1.
+Evaluation Process
+To validate the proposed mutation operators, we ap-
+ply them to ML/SL models. We separate the evalu-
+ation process into two parts base and extension, ac-
+cording to the ability of ML/SL. We apply the first-
+order mutants (FOMs) 22 to ML/SL models to gen-
+erate a mutant, which means we generate a mutant
+by using a mutation operator only once.
+5.1.1. Base Case
+In the base case, we examine the simulation results
+of the original models and the SimSched models and
+their mutants. An original model M is an ML/SL
+model scheduled by Stateflow scheduler; A Sim-
+Sched model M ′ is an original model scheduled
+by SimSched; The mutants (Mµ or M ′
+µ) are ei-
+ther original model or SimSched models mutated by
+one of our mutation operators. Figure 28 shows the
+schematic diagram of our mutants generation pro-
+cess. We use the simulation result of M as a com-
+parison baseline, and then we compare the baseline
+with every other simulation result of Mµ, and M ′
+µ.
+We examine the comparison result to see if the re-
+sult reaches a verdict failure during model simula-
+tion. We say a mutant is killed if a verdict of failure
+is reached.
+
+In2
+Out1
+A
+Task A
+Out1
+In2
+In1
+Qut3
+Task B1
+Task_B2A
+2
+DSRA
+2J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 28. Schematic diagram of the model mutants genera-
+tion process.
+Fig. 29. Simple evaluation example scheduled by Stateflow
+scheduler.
+We use a simple example shown in Figure 29 to
+explain the base case evaluation process. This ex-
+ample is an original model. We replace the State-
+flow scheduler with a SimSched scheduler to form
+a SimSched model. We generate mutants for both
+the original and SimSched models by a specific mu-
+tation operator, e.g., mDTPER, to decrease the task
+period. Then we run the simulation for both mutants
+and analyzed the results to see if there is any errors.
+If the simulation result of Mµ or M ′
+µ is different
+from the original model and shows a verdict failure,
+then we say the mutant is killed.
+In this example,
+we have three runnables
+R1,R2,R3 and they are mapped to two tasks T1,T2.
+R1 is mapped to T1 and R2,R3 are mapped to T2. The
+period of T1 is 3ms and The period of T2 is 6ms.
+The execution time of each runnable is 1ms. The
+simulation result of the M is shown in Figure 30
+and it shows each runnable output is a rising non-
+interlaced polyline. We apply the mDTPER muta-
+tion operator as decreasing 1ms to both the origi-
+nal model and SimSched model to generate mutants.
+The task T1 in the mutants has a period of 2ms. The
+simulation result of these simulations is shown in
+Figure 31 and Figure 32. The simulation result of
+M ′
+µ is different from the result of M , and it shows
+the output of R2 and R3 are two rising interlaced
+polylines because SimSched can simulate the exe-
+cution time and preemption. T1 preempts T2 in the
+SimSched mutant model to yield an alternative ex-
+ecution trace, and we say a verdict fail is reached.
+However, the simulation result of Mµ is similar to
+the result of M . Thus, the mDTPER mutant is killed
+to the M ′
+µ and is alive to the Mµ. We can not apply
+this means to all mutation operators due to the nature
+of ML/SL. We combine this method and the follow-
+ing method to evaluate the mutation operators.
+Fig. 30. M simulation result.
+Fig. 31. Mµ simulation result.
+Fig. 32. M ′µ simulation result
+5.1.2. Extension
+To evaluate the rest of the mutation operators, we
+implement a mutation generator with additional
+functionalities to assist the validation process. One
+feature is to check the mutant model’s schedulabil-
+ity at the given set of tasks configuration to decide if
+all task deadlines are met. The other function is to
+
+M'
+M
+SimSched
+Mutate
+Mutate
+M
+M'
+μ
+n1 ms Clock
+A
+R1()
+R3()
+R2()
+Sall(
+R2
+R1
+call()
+R3
+R10
+R2()
+R30
+Scheduler90
+R1
+R2
+R3
+80
+70
+60
+50
+40
+30
+20
+10
+0
+-10
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06R1
+R2
+120
+R3
+100
+80
+60
+40
+20
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06R1
+60
+R2
+R3
+50
+40
+30
+20
+10
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Offset=0J. Chen et al. / Mutation Operators for Simulink Models
+check the data access sequence. If there is a DataS-
+tore block in the mutated model, every read or write
+to this DataStore block is recorded. Then we use
+this mutated model data access sequence to com-
+pare with the original model data access sequence.
+The mutation generator is implemented as a Matlab
+script written in m-file.
+The validation process takes a Stateflow sched-
+uled ML/SL model and a test specification as input.
+The test specification specifies which mutation op-
+erator to use. A mutant generator applies the speci-
+fied mutation operator to the ML/SL model via Sim-
+Sched and generates a mutant. The mutant genera-
+tor then executes the simulation both for the original
+model and the mutated model using the additional
+functionalities to analyze the simulation. If the anal-
+ysis shows at least one task misses its deadline in a
+mutated model, then we say a mutant is killed. Or
+at least one variable comparison result of the DataS-
+tore access sequence is unmatching, and then we say
+a mutant is killed; otherwise, we report the mutant
+is benign.
+Fig. 33. A simple example of using Model Scheduler to
+schedule AUTOSAR SW-Cs.
+We use an example shown in Figure 33 to ex-
+plain the validation process. It has three runnables
+and is mapped to two tasks, R1 map to T1, R2, and
+R3 map to T2. The period of task T1 is 10ms, and
+T2 is 20ms, every runnable’s execution time is 3ms.
+There is a DataStore block named A as a shared vari-
+able in this example model. If we apply the period
+mutation operator mDTPER ρi − δ where i − 1 and
+δ = 6 to this model to decrease the period of T1 and
+generate a mutant, run it. The analysis result shows
+the T2 missed deadline, then we say this mutant is
+killed. If we apply the execution time mutation op-
+erator mITET ci + δ where i = 1 and δ = 3 to this
+model to increase the execution time for T1 and gen-
+erate a mutant. The DataStore access sequence of
+the original model is a pattern of WRWR where W
+represents a write to the shared variable, and R rep-
+resents a read to the shared variable. The mutant
+generates a different sequence, which is WRWWR.
+It is because the T1 has a longer execution time than
+the original model, and it preempts T2 during the ex-
+ecution of T2. Hence, there is one more W in the
+DataStore access sequence.
+5.2.
+Experiments
+We employ two examples to demonstrate the use of
+our mutation testing framework. We first explain the
+two examples in detail. We then apply the mutation
+operators to the two models scheduled by both the
+Stateflow scheduler and SimSched.
+Fig. 34.
+The three-servo example adapted from 12 with
+Stateflow scheduler.
+5.2.1. Three Servos Model
+We adapt an example from the TrueTime 21 exam-
+ple library, which shows a possible implementation
+of a three-servo PID control system. The example
+is shown in Figure 34 with a Stateflow scheduler.
+In this example, three DC servos are modeled by a
+continuous-time system, and three PID controllers
+are implemented as three subsystems. We map three
+controller subsystems to three runnables R1, R2, and
+R3 then they are mapped to tasks T1, T2, and T3. The
+task periods are T1=4 , T2 = 5 and T3 = 6 ms re-
+spectively. Each task has the same execution time as
+
+Scheduler
+Runnable(period, execution time, priority) Task()
+function()
+function()
+function()
+R1(10ms,3ms,2)j
+Task(1)
+R2( 20ms, 3ms, 1
+Task(2)
+R3( 20ms, 5ms, 1
+Task(2
+function
+irv2
+irv1
+irv2
+SimSched
+irv3 
+Aader
+Runnable1_subsystem
+Runnable2_subsystem
+Runnable3_subsystem1 ms Clock
+callo
+R10
+R2()
+R30
+call(
+Temporal Logic
+Scheduler
+functionO
+1000
+u
+s2+s
+DCServo1
+PID1
+functionO
+1000
+u
+2+s
+DCServo2
+PID2
+function()
+1000
+u
+s?+s
+DCServo3
+PID3J. Chen et al. / Mutation Operators for Simulink Models
+1ms. Task settings are shown in Table 7. The simula-
+tion result is shown in Figure 35 based on the above
+task settings. The three graphs show the output of
+the motors using the three PID controllers when the
+corresponding task parameters are assigned accord-
+ingly. In the graph, the square wave is the reference
+input signal for the motors, where the computation
+delays are not taken into account. Three PID con-
+trollers are all smooth output signals as expected.
+Table 7. Three Servo example settings.
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+4
+1
+3
+R1
+T2
+5
+1
+2
+R2
+T3
+6
+1
+1
+R3
+Fig. 35.
+The three servos example output with Stateflow
+scheduler.
+We replace the Stateflow scheduler with the Sim-
+Sched scheduler, and the updated example is shown
+in Figure 36. In this example, three DC servos have
+the same task setting as the Stateflow scheduler ex-
+ample. Each runnable has the same execution time
+as 1ms. The simulation result is the same as the
+Stateflow scheduler example based on the above task
+settings. There is no deadline missing for any task,
+so the simulation result shows every task has smooth
+control. Figure 37 shows the task active chart gen-
+erated by SimSched. Every task has been executed
+within its own deadline.
+Fig. 36. The three servos example adapted from 12.
+Fig. 37. The three servos example task active chart gener-
+ated by SimSched.
+Fig. 38. The adjusted Stateflow scheduler for mITO muta-
+tion operator to increase o f fset as 1ms for DCServo1.
+
+2
+DCServo1
+1
+SignalGenerator
+0
+-1
+-2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+2
+DCServo2
+1
+SignalGenerator
+0
+-1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9Scheduler
+Runnable(period, execution time, priority) Task()
+E
+R1(4ms, 2ms, 3) Task(1)
+R2( 5ms, 2ms, 2
+Task(2)
+R3( 6ms, 2ms, 1
+ Task(3
+SimSched
+function(
+○○
+1000
+u
+s2+s
+DCServo1
+PID1
+function()
+1000
+u
+2+s
+y
+DCServo2
+PID2
+function()
+1000
+u
+32+s
+DCServo3
+PID30.5 
+o
+0
+0. 01
+0.02 
+0.03 
+to0
+0. 05
+0.06 
+0.5
+0
+0.01
+0.02 
+0.03 
+to0
+0.05 
+0.06 
+1F
+0.5 
+0
+0.01
+0.02 
+0.03 
+to0
+0.05 
+0.06 Scheduler
+on at(1,tick): Period_4_ms;
+du: on every(5,tick) : Rate5ms;
+du: on every(6,tick) : Rate6ms;
+'Sched_4_MS
+Periodicl
+on every(4,tick) : Period_4_ms;
+Sched 5 MS
+Periodicl
+en: Period_5_ms;
+Rate5ms
+Sched 6 MS
+Periodic/
+en: Period_6_ms;
+Rate6msJ. Chen et al. / Mutation Operators for Simulink Models
+Next step, we apply mITO, mDTO, mITPER,
+mDTPER, mARPREC, mRRPREC, mITJ, mDTJ
+mutation operators to both Stateflow scheduler and
+SimShced examples to generate two versions of mu-
+tants with the same mutation operators.
+To ap-
+ply some of the mutation operators to evaluate the
+Stateflow scheduler, we need to adjust the State-
+flow scheduler so that it can be used on the gen-
+erated mutants. Figure 38 shows an example that
+is adjusted for the Offset mutation operator. This
+example uses a temporal logic operator at in the
+state to set the Offset parameter to generate a mu-
+tant for PID1 which runs at the period of 4ms in
+this example. This mutant increases Offset as 1ms
+for DCServo1 controlled by PID1. The mutant of
+the SimSched version can be easily generated by our
+model scheduler SimSiched.
+Fig. 39. The Stateflow scheduled three-servo example task
+active chart after applying mITO mutation operator to in-
+crease o f fset as 1ms for DCServo1.
+We run simulations for both versions of the mu-
+tants generated by Offset mutation operator. Both
+mutant versions’ output of three servos is the same
+as shown in Figure 35. The only difference occurs
+at the beginning of the simulation but it does not af-
+fect the smooth control of DCServos. We can see
+the difference from the following comparison. Fig-
+ure 39 shows the Stateflow scheduled task active
+chart after applying mITO mutation operator to in-
+crease offset as 1ms for DCServo1. Before apply-
+ing the mutation operator, every task is released at
+time 0. After applying the offset mutation opera-
+tor, Task 1 is delayed by 1ms shown on the top of
+the figure. Task 2 and Task 3 are both released at
+time 0. Figure 40 shows the SimSched scheduled
+three servos example task active chart after applying
+mITO mutation operator to increase offset as 1ms for
+DCServo1. There are three output signals represent-
+ing three tasks from top to bottom T1, T2, and T3.
+As T1 has a 1ms offset, Task 2 is executed first as
+shown in the figure the second line starts at time 0.
+Because the SimSched scheduler has the execution
+time parameter, Task 2 is executed at time 1 and Task
+3 at time 2 respectively.
+Fig. 40. The SimSched scheduled three servos example task
+active chart after applying mITO mutation operator to in-
+crease Offset as 1ms for DCServo1.
+We use a similar approach to apply Period muta-
+tion operator to the three-servo example and gener-
+ate mutants for both the Stateflow scheduler model
+and SimSched model. We use two mutation con-
+figurations to show the similarities and differences
+between the two schedulers. The first configuration
+is [1,5,6]. It means Task 1 has a period of 1ms and
+the period of Task 2 and Task 3 keep the same as
+5ms and 6ms, respectively.
+Figure 41 shows the
+Stateflow scheduler mutant simulation result. Be-
+cause Task 1 has a 1ms period, it has too many times
+of calculations, and the output value is out of the
+chart. Task 2 and Task 3 keep the same output as be-
+fore. Figure 42 shows the SimSched mutant simula-
+tion result. The output of Task 1 is the same as the
+Stateflow scheduler mutant. However, Task 2 and
+Task 3 are different from the Stateflow one. Because
+the SimSched takes the execution time into account,
+Task 1 has the highest priority and has an execution
+time of 1ms, and Task 1 always runs during the sim-
+ulation. Task 2 and Task 3 always are preempted by
+Task 1 because they have a lower priority than Task
+1.
+Fig. 41. The Stateflow scheduled three servos example out-
+put after applying mDTPER mutation operator to decrease
+period as 1ms for DCServo1.
+
+PID1  4 ms 1ms offset
+0.01
+0.02
+0.03
+0.04 
+0.05 
+0.06
+PID2 5 ms no offset
+1
+0.5 
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05 
+0
+0.06
+PID3 6 ms no offset
+1
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05 
+0.061
+0.5,
+0 
+0
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+1 
+0.5 
+0
+0.01 
+0.02 
+0.03
+to'0
+0.05
+0.06
+1 
+0.5 
+0 
+0
+0.01
+0.02 
+0.03 
+to'0
+0.05
+0.062
+DCServo1
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7 
+0.8
+0.9
+2
+DCServo2
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2 
+0.3
+0.4 
+0.5 
+0.6
+0.7
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+-2
+0
+0.1
+0.2
+0.3
+0.4 
+0.5
+0.6
+0.7
+0.8
+0.9J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 42. The SimSched scheduled three servos example out-
+put after applying mDTPER mutation operator to decrease
+period as 1ms for DCServo1.
+The second configuration is [13,5,6]. It means
+Task 1 has a period of 13ms and the period of Task
+2 and Task 3 keep the same as 5ms and 6ms, re-
+spectively. Figure 43 shows the Stasflow scheduler
+mutant simulation result and the SimSched mutant
+has the same output as shown in the figure. Because
+Task 1 has a 13ms period, it has fewer computations
+than the original model. Although the output behav-
+ior looks like Task 1 misses its deadline in the figure,
+every execution of Task 1 meets its deadline and it
+is executed as scheduled.
+Fig. 43. The Stateflow scheduler three servos example out-
+put after applying mITPER mutation operator to increase
+period as 13ms for DCServo1.
+We
+generate
+mutants
+for
+mARPREC
+and
+mRRPREC operators by setting each parallel state’s
+execution order in the Stateflow scheduler model
+and configuring the parameters and connections for
+the SimSched model. The two mutants’ simulation
+results are the same as the original model, except
+that each task’s execution order is different from the
+original model.
+We generate mutants for mITJ and mDTJ op-
+erators by adapting the Stateflow scheduler in the
+model and configuring the parameters in the Sim-
+Sched model. We set the configuration as [1,0,0].
+It means only Task 1 has a jitter as 1ms. Figure 44
+shows the SimSched scheduler three servos example
+task active chart after applying mITJ mutation oper-
+ator to increase jitter as 1ms for DCServo1. Be-
+cause Task 1 has 1ms jitter, Task 2 is executed first
+then Task1 and Task3.
+Fig. 44. The SimSched scheduler three servos example task
+active chart after applying mITJ mutation operator to in-
+crease jitter as 1ms for DCServo1.
+We only apply the execution time operator to the
+SimSched model due to the lack of support for the
+Stateflow scheduler. Figure 45 shows the effect out-
+put of mITET mutation operator. We set c1 = 3ms
+using the mITET mutation operator for Task 1 to
+generate a mutant. The output of Task 3, shown as
+DCservo 3 at the bottom of the figure, is a curly
+wave. It is an unstable control due to the preemp-
+tion by T1 and T3 missing its deadline. Figure 46
+shows the task preemption effect. Task 1 takes 3ms
+to execute and Task 2 takes 1ms to execute. After
+the execution of Task 1 and Task 2, Task 3 should be
+executed; however, it is the time that Task 1 is sched-
+uled to run. Task 1 has a higher priority, so Task 3 is
+preempted by Task 1. The first instance of Task 3 is
+executed at 19ms so Task 3 does not have a smooth
+control signal output as the other tasks. Although
+some task preemptions occur in Task 2, it does not
+miss enough deadlines to significantly affect the out-
+put. Task 2 still has a smooth output signal as shown
+in Figure 45 as the second chart.
+Fig. 45. The SimSched scheduler three servos example sig-
+nal output after applying mITET mutation operator to in-
+crease executiontime to 3ms for DCServo1.
+
+2
+DCServo1
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7 
+0.8
+0.9
+2
+DCServo2
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2 
+0.3 
+0.4
+0.5
+0.6
+0.7 
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+-2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.92
+DCServo1
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+2
+DCServo2
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2 
+0.3
+0.4 
+0.5 
+0.6
+0.7
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+-2
+0
+0.1
+0.2
+0.3
+0.4 
+0.5
+0.6
+0.7
+0.8
+0.9DCServo1
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+DCServo2
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+DCServo3
+1
+0.5
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05DCServo1
+2
+0
+-1
+2
+0.1
+0.2 
+0.3 
+0.4 
+0.5 
+0.6 
+0.7 
+0.8
+0.9 
+DCServo2
+2
+-1
+。
+0.1
+0.2
+0.3 
+0.4 
+0.5 
+0.6 
+0.7 
+0.8
+0.9 
+DCServo3
+2 
+1
+0
+-1
+-2
+0.1 
+0.2 
+0.3
+0.4 
+0.5
+0.6
+0.7
+0.8 
+0.9J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 46. The SimSched scheduler three servos example task
+active chart after applying mITET mutation operator to in-
+crease executiontime to 3ms for DCServo1.
+5.2.2. Throttle Position Control Model
+We adopt an AUTOSAR software component
+Simulink model from Mathworks shown in Figure
+47. It implements a throttle position control sys-
+tem for an automobile and contains three sensors,
+one monitor, one controller, and one actuator. They
+are implemented as six subsystems and mapped to
+six runnables TPSSecondary, Monitor, Controller,
+Actuator, APPSnsr and TPSPrimary then they are
+mapped to tasks T1, and T2. The task periods are
+T1=5ms and T2 = 10 ms respectively. Each runnable
+has the same execution time of 1ms. Task settings
+are shown in Table 8. This example uses seven Data-
+StoreMemory blocks to access the shared resources.
+Fig. 47. Throttle position control Simulink model contains
+six runnables.
+Table 8. Throttle control example settings.
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T2
+10
+1
+1
+TPSPrimary
+T1
+5
+1
+2
+TPSSecondary
+T1
+5
+1
+2
+Monitor
+T1
+5
+1
+2
+Controller
+T1
+5
+1
+2
+Actuator
+T2
+10
+1
+1
+APPSnsr
+Figure 48 shows the simulation result, which is
+generated by a Stateflow scheduler. The square wave
+in the figure is the simulated pedal input, and the
+curly wave is the output of the throttle body, repre-
+senting the current throttle position. The Stateflow
+scheduler simulates the throttle control controller’s
+process well and simulates the entire control pro-
+cess.
+Fig. 48. The simulated throttle position of the throttle posi-
+tion control model scheduled by the Stateflow scheduler.
+Figure 49 shows the runnable active chart sched-
+uled by the Stateflow scheduler. All runnables are
+scheduled and executed at time 0. Runnable TP-
+SPrimary and APPSensor are mapped to T2 and they
+are scheduled and executed every 10ms and the top
+and bottom chars shown in the figure are their active
+charts. The active charts of runnable TPPSSendary,
+Monitor, Controller, and Actuator are the four charts
+in the middle of the figure. They are scheduled and
+executed every 5ms.
+Fig. 49. The runnable active chart of the throttle position
+control model scheduled by the Stateflow scheduler.
+We apply SimSched to the Stateflow scheduler
+model, and we can get a similar simulation result as
+the Stateflow scheduler example based on the above
+task settings. Figure 50 shows the task active chart
+generated by SimSched. Every task has been exe-
+cuted within its own deadline. Both task T1 and T2
+are scheduled at time 0 but only T1 is executed at
+
+DCServo1
+1
+0.5
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05 
+DCServo2
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+DCServo3
+0.5 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.054
+TPSSecondaryRun5ms
+MonitorRun5ms
+ControllerRun5ms
+function()
+function()
+function(
+汇
+ThrottlePositionSensorSecondary
+ThrottlePositionMonitor
+Controller
+6
+APPSnsrRunl
+ActuatorRun5ms
+TPSPrimaryRuniOms
+function()
+function()
+function()
+口
+APPHwlOValueread
+ThrCmdHwlOValuewrite1
+ThrottlePositionSensorPrimary
+AccelerationPedalPositionSensor
+ThrottlePositionActuator
+APP_HwlO_Value
+ThrCmd HwlO ValueThrottle Pos
+0.7
+Simulated Pedal Input
+0.6
+Throttle Pos
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.5
+1.5TPSPrimary
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+TPSSecondary
+1
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03 
+0.035
+0.04 
+0.045
+0.05
+Monitor
+1
+0.5 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045 
+0.05
+Controller
+1
+0.5
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Actuator
+0.5
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04
+0
+0.045
+0.05
+APPSensor
+3
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+s0'0
+0J. Chen et al. / Mutation Operators for Simulink Models
+time 0 due to its higher priority. T1 takes up 4ms to
+run. After the first instance of T1 is finished, T2 is
+executed. The second instance of T1 arrives at 5ms,
+which is during the middle of the execution of T2. T2
+is preempted by T1 and resumes at the completion of
+the second instance of T1.
+Fig. 50. The task level active chart of the throttle position
+control model scheduled by the SimSched scheduler.
+Figure 51 shows the runnable level active chart
+of the throttle position control model scheduled by
+the SimSched scheduler. From this figure, we can
+clearly see the activity of each runnable.
+It ex-
+actly shows the execution order of each runnable.
+Runnable TPPSSendary, Monitor, Controller, and
+Actuator are executed one after another followed by
+TPSPrimary. Runnable APPSensor is executed af-
+ter the second instance of T1 and it is the preemption
+point of T2.
+Fig. 51. The runnable active chart of the throttle position
+control model scheduled by the SimSched scheduler.
+We use the same means applied to three servos
+example to apply it to the throttle position control
+model. We adopt the Stateflow schedule and replace
+it with SimSched to generate mutants for both the
+Stateflow scheduler and SimSched for the experi-
+ments.
+Figure 52 shows the runnable active chart of the
+Throttle Position Control model scheduled by Sim-
+Sched after applying mTIO mutation operator as in-
+creasing 2ms offset for T1. The runnable Actuator
+active chart shown in the second bottom chart in the
+figure is missing its first execution. T1 takes 4ms to
+run, and it also has 2ms offset, so the total execution
+time of T1 exceeds its period 5ms. On the other hand,
+the mutant generated by the Stateflow scheduler can
+not simulate this overrun situation due to the lack of
+execution time simulation support.
+Fig. 52. The runnable active chart of Throttle Position Con-
+trol model scheduled by SimSched after applying mTIO
+mutation operator as increasing 2ms offset for T1.
+Figure 53 shows the simulated throttle position
+of the throttle position control model scheduled by
+SimSched after applying mITPER mutation opera-
+tor for T1 at 100ms. The Stateflow scheduler also
+can output the same figure. The mITPER mutation
+operator can reduce the computation times of a task
+at the same amount of time, which results in unsta-
+ble control as shown in the figure.
+Fig. 53. The simulated throttle position of the throttle po-
+sition control model scheduled by the Stateflow scheduler
+after applying mITPER mutation operator.
+Figure 54 shows the simulated throttle position
+of the throttle position control model scheduled by
+SimSched after applying mDTPER mutation opera-
+tor for T1 at 4ms. The mDTPER can result in no
+output signal for T2 because a higher rate task in-
+creases the computation times, and the lower rate
+task does not get an execution. In this example, we
+set ρ1 = 4ms using the mDTPER mutation operator,
+
+Task1
+1
+0.5
+oE
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03 
+0.035
+0.04 
+0.045
+0.05
+Task2
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05TPsPrimary
+0.5 
+10
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+TPSSecondary
+0.5
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Monitor
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Controller
+0.5 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+Actuator
+0.5
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0
+0.05
+APPSnsr
+0.5 /
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05TPsPrimary
+1
+0.5
+10
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+TPSSecondary
+1 
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Monitor
+1
+0.5
+0 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+Controller
+1 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+Actuator
+1 F
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+APPSnsr
+0. E
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05Throttle Pos
+Simulated Pedal Input
+Throttle Pos
+0.8
+0.6
+0.4
+0.2
+0
+0.5
+1.5J. Chen et al. / Mutation Operators for Simulink Models
+then the T2 is always preempted by T1 and does not
+have a chance to execute.
+Fig. 54. The simulated throttle position of the throttle po-
+sition control model scheduled by SimSched after applying
+mDTPER mutation operator.
+We only apply the execution time operator to
+SimSched models. We set c1 = 5ms using mITET
+mutation operator for T1 to generate a mutant. This
+mutant just outputs the same throttle position as
+shown in Figure 54. Since T1 has increased its exe-
+cution time by 1ms, it just takes up all the time slots
+in its period. T2 is preempted by T1 during the simu-
+lation process.
+We apply mARPREC and mRRPREC mutation
+operators to this throttle position control exam-
+ple. First, there is no precedence between runnable
+APPSnsr and TPSPrimary, and we use a mARPREC
+mutation operator to add precedence γAPPSnsr to
+precrTPSPrimary to generate mutants for both sched-
+ulers.
+Runnable Controller consumes the values
+produced by APPSnsr and TPSPrimary to calcu-
+late the throttle percent value for the throttle actu-
+ator. The changes in the simulation results of both
+mutants are trivial. Figure 55 shows the simulation
+result comparison between the original model and
+the SimSched mutant.
+Second, there is precedence between Controller
+and Actuator, and Controller is executed before
+Actuator. We remove the precedence from the pair
+of runnables, so the Actuator (destination) runs be-
+fore Controller (source), which changes the data de-
+pendency and delays the data. The difference in sim-
+ulation is similar to Figure 55.
+Fig. 55. The difference of simulation result between the
+original model and the mutant with mARPREC mutation
+operator scheduled by SimSichde.
+We apply mDSM, mUDSM, mRDSM, mRSM,
+mRMSMR, and mRSMR mutation operators to this
+example and generate accordingly mutants for both
+the Stateflow scheduler and SimSched.
+Inter-
+estingly, since the task time properties have not
+changed for the shared memory mutation operators,
+the simulation results are consistent between the two
+types of mutants. For example, we apply the mDSM
+mutation operator to variable ThrCmdPercentValus
+and set the variable to a new constant value. The
+simulation result of both types of mutants is the
+same shown in Figure 56. Since this variable is an
+input to a Lookup table, there is always an output
+that matches the input value and yields a correspond-
+ing value to the output. The input value is constant,
+so the throttle position’s output is a smooth curve.
+Fig. 56. The throttle position output after applying mDSM
+mutation operator to variable ThrCmdPercentValus sched-
+uled.
+5.3.
+Evaluation Result
+We apply the evaluation process to the example
+models to investigate the mutation operator’s effec-
+tiveness to kill mutants.
+Table 9 and 10 summa-
+rize the results of the efficacy of mutation operators,
+
+Throttle Pos
+Simulated Pedal Input
+0.6
+Throttle Pos
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.5
+1.5ThrottlePos(Run2:TPCWithTaskAndDSM_CaseStudy_Stateflow)
+ThrottlePos (Run5:TPCWithTaskAndDSM_CaseStudy_SimSched)
+Tolerance
+0.6 
+0.5
+0.4 
+0.3 
+0.2
+0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5Throttle Pos
+0.7
+Simulated Pedal Input
+Throttle Pos
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.5
+1.5J. Chen et al. / Mutation Operators for Simulink Models
+where each row provides the number of mutants and
+the mutation score of our mutation operators. The
+mutation score is a measure that gives the percent-
+age of killed mutants with the total number of muta-
+tions.
+Table 9. Mutation analysis of mutation operators for three ser-
+vos example.
+Stateflow Scheduler
+SimSched
+Operator
+Mutants
+Kills
+Mutants
+Kills
+Offset
+36
+24
+36
+27
+Period
+39
+25
+39
+25
+Execution
+N/A
+N/A
+36
+31
+Time
+Precedence
+5
+0
+5
+0
+Priority
+18
+0
+18
+0
+Jitter
+36
+24
+36
+27
+Mutation
+54.48%
+64.71%
+Score
+For the three servo example, we generate 134
+mutants for the Stateflow scheduler models and 170
+mutants for the SimSched models. We achieve a
+mutation score of 54.48% for the Stateflow sched-
+uler model and 64.71% for the SimSched models.
+Evaluation results show that the time-related muta-
+tion operators have the most effect on the mutation
+testing, such as the Offset, Period, Execution Time,
+and Jitter mutation operator. We observe that the
+Precedence and Priority mutation operators kill zero
+mutant because, in this example, each controller in-
+dividually controls a motor. There is no connection
+between them, so the change of precedence and pri-
+ority does not cause any simulation changes. How-
+ever, the three controllers run on a single CPU, and
+one task execution time’s length affects other tasks.
+We also observe the Offset mutation operator only
+affects each task’s initial execution, and tasks miss
+the deadline. Still, each mutant’s simulation results
+show each controller can have stable control of each
+servo.
+Table 10. Mutation analysis of mutation operators for throttle
+position control example.
+Stateflow Scheduler
+SimSched
+Operator
+Mutants
+Kills
+Mutants
+Kills
+Offset
+19
+7
+19
+15
+Period
+30
+6
+30
+19
+Execution
+N/A
+N/A
+34
+34
+Time
+Precedence
+23
+10
+23
+10
+Priority
+11
+0
+11
+0
+Jitter
+19
+12
+19
+15
+Shared
+72
+46
+72
+46
+Memory
+Mutation
+49.39%
+70.2%
+Score
+For the throttle position control example, we
+generate 164 mutants for the Stateflow scheduler
+models and 198 mutants for the SimSched models.
+We achieve a mutation score of 49.39% for the State-
+flow scheduler model and 70.2% for the SimSched
+models. Evaluation results are similar to the previ-
+ous example. The time-related mutation operators
+have the most effective for mutation testing.
+We
+observe that the Shared Memory mutation operators
+have the same kills for both Mµ and M ′
+µ. In this
+example, task T2 has two Runnable TPSPrimary and
+APPSnsr each has a shared variable to update at each
+execution, and there is no direct relation between
+them. Though SimSched can simulate the preemp-
+tion of T2 to interrupt its execution, shared memory
+mutants’ model behaviors are the same for both Mµ
+and M ′
+µ.
+From the above two examples, we can see the
+mutation operators are application-dependent, and
+SimSched can achieve a higher mutation score at
+the time-related mutation operators. For example,
+the Precedence mutation operator kills zero mutants
+for the three servo example, but it kills ten mutants
+for the throttle position control example for both
+Stateflow Scheduler and SimSched. The three-servo
+example does not require any precedence at all;
+each task only controls itself. However, the throt-
+tle position control example requires precedence. A
+runnable consumes data from a previous runnable
+execution. If we alter the precedence, the execu-
+tion order is different from the original model ex-
+
+J. Chen et al. / Mutation Operators for Simulink Models
+ecution; it produces a different data flow. For exam-
+ple, in the Period mutation operator both Stateflow
+scheduler and SimSched kill the same mutants for
+the three serve example, but SimSched kills more
+mutants than the Stateflow Scheduler in the throttle
+position control example. Because the throttle po-
+sition control example has four runnables in T1 and
+each runnable has 1ms execution time. We use a
+mDTPER to T1 and generate a mutant that the pe-
+riod of T1 is 4ms instead of 5ms, then T1 will occupy
+all the execution time slots, and T2 will not be ex-
+ecuted for the SimSched. However, the Stateflow
+Scheduler does not consider the execution time, so
+both T1 and T2 are executed as scheduled.
+6.
+Related Work
+Our work aligns with the MBMT, and it has been
+applied to various models.
+Trakhtenbrot 41 pro-
+poses mutation testing on statechart-based models
+for reactive systems. This approach mainly deals
+with the conformance between specific semantics
+of statechart models and the model’s implementa-
+tion.
+Mutation testing has been applied to fea-
+ture models to test software product lines 20. The
+feature models represent the variability of software
+product lines and configurable systems.
+El-Fakih
+et al.14 develop a technique of mutation-based test
+case generation toward extended finite state ma-
+chines (EFSM) that examines the EFSM under test
+agrees to user-defined faults. Belli et al.6 have sur-
+veyed MBMT approach a great deal and detailed the
+approach applied to graph-based models, including
+directed graphs, event sequence graphs, finite-state
+machines, and statecharts. A recent mutation testing
+survey31 presents up-to-date advanced approaches
+that use mutants to support software engineering ac-
+tivities to model artifacts.
+Our work also fits in the timed system test-
+ing, which requires a real-time environment. Re-
+searchers 3,28,29 have utilized the most studied for-
+malisms TA to inject faults to the timed system and
+reveal that the time-related errors are unable to find
+by using randomly generated test suites.
+Nilsson
+et al.28 first proposed a set of extended TA muta-
+tion operators based on TA with Tasks (TAT) 30 to
+test real-time systems which depend on the execu-
+tion time and execution order of individual tasks.
+The mutation operators are interested in the time-
+liness of a task to meet its deadlines. Aichernig et
+al.3 propose a mutation testing framework for timed
+systems, where they define eight mutation operators
+to mutate the model and its mutants are expressed
+as a variant of TA in Uppaal specification format.
+The authors also develop a mutation-based test case
+generation framework for real-time, where they use
+symbolic bounded model checking techniques and
+incremental solving. Cornaglia et al.11 presents an
+automated framework MODELTime that facilitates
+the study of target platform-dependent timing dur-
+ing the model-based development of embedded ap-
+plications using MATLAB/Simulink simulations.
+ML/SL is one of the most popular formalisms to
+model and simulate embedded systems, and many
+researchers have explored the various type of mu-
+tation operators applied to ML/SL models. Hanh
+et al.18 propose a set of mutation operators based
+on investigating common faults in ML/SL models
+to validate test suites and present a process of muta-
+tion testing for ML/SL models. They provide twelve
+mutation operators and divide them into five cate-
+gories. Stephan et al.38 utilize the mutation testing
+technique to compare model-clone detection tools
+for ML/SL models.
+They present a taxonomy of
+ML/SL and prose a set of structural mutation op-
+erators based on three clone types. The mutation
+operators are used to evaluate the model-clone de-
+tectors. Using the mutation-based technique to gen-
+erate test cases for ML/SL models automatically has
+been studied 8,19.
+They can effectively generate
+small sets of test cases that achieve high coverage
+on a collection of Simulink models from the auto-
+motive domain. A recent work SLEMI 10 has ap-
+plied mutation techniques to the Simulink compiler
+and uses tools to generate mutants of the seed model
+and found 9 confirmed bugs in Simulink models.
+Our work intends to exploit mutation analysis
+to identify potential time-related errors in ML/SL
+models.
+Roy and Cordy 33,34 first propose using
+mutation analysis to assist the evaluation of soft-
+ware clone detection tools. They develop a frame-
+work for testing code-clone detectors based on mu-
+
+J. Chen et al. / Mutation Operators for Simulink Models
+tation. Stephan et al.37,36 proposed a framework that
+can objectively and quantitatively evaluate and com-
+pare model-clone detectors using mutation analysis.
+Their work is based on a structural mutation method
+for ML/SL model mutation. Our mutation operators
+are based on a timed system task model, whereas,
+there are no relevant existing studies that directly
+integrated the ML/SL models in the timed systems
+in the MIL phase; thus, we carry out the work pre-
+sented in this paper.
+Co-simulation16 is a widely used technique in
+model-based testing to verify as much of the system
+functionality, among subsystems, as possible. Com-
+posing the simulations of sub-simulators can achieve
+a joint simulation of a coupled system. Many differ-
+ent languages and tools are used for other purposes
+in the model-based design domain, either designing
+continuous plants or discrete controllers.
+A rela-
+tively recent open standard functional mock-up in-
+terface (FMI) is developed for exchange simulation
+models in a standardized format, including support
+for co-simulation. Gomes et al.15 propose an ap-
+proach to facilitate the implementation of the Func-
+tional Mock-up Interface standard.
+They use the
+MBT methodology to evaluate the tools that export
+Functional Mock-up Units (FMUs). Hence, they can
+root out the ambiguities and improve conformance
+to the FMI standard. Garro et al.7 employs FMI to
+perform co-simulation to verify the system require-
+ments based on the FOrmal Requirements Model-
+ing Language and the Modelica language.
+Zafar
+et al.42 present a systematic tool-supported MBT
+workflow to facilitate the simulation-based testing
+process of an embedded system. The workflow ex-
+pends from the requirements phase, and generation
+of executable test scripts, to the execution of gener-
+ated test scripts on simulation levels.
+7.
+Conclusion and future work
+In this paper, we proposed a set of timed muta-
+tion operators for the ML/SL model that is primar-
+ily intended to integrate the timed task model in the
+ML/SL model to better support MIL simulation us-
+ing mutation analysis. Moreover, testing at an ear-
+lier stage during the development process reduces
+development costs since earlier changes and fixing
+errors are much more manageable. We introduce a
+timed task model and present a set of mutation op-
+erators for the ML/SL based on this task model. We
+implement a mutation analysis framework that can
+apply mutation operators to the simple ML/SL mod-
+els. We demonstrate the approach on several ML/SL
+models. The results validate that mutation analysis
+can reveal time-related faults. We intend to automate
+the mutation testing process for the ML/SL environ-
+ment and improve the mutation operators to expose
+defects in the future. We will further validate our
+mutation analysis method to more industrial com-
+plex ML/SL model sets.
+Acknowledgments
+This work was supported in part by the Natural Sci-
+ences and Engineering Research Council of Canada
+(NSERC), as part of the NECSIS Automotive Part-
+nership with General Motors, IBM Canada, and Ma-
+lina Software Corp.
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+

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+arXiv:2301.01062v1  [math.AT]  3 Jan 2023
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+OSCAR RANDAL-WILLIAMS
+Abstract. We describe the ring structure of the rational cohomology of the
+Torelli groups of the manifolds #gSn × Sn in a stable range, for 2n ≥ 6.
+Some of our results are also valid for 2n = 2, where they are closely related to
+unpublished results of Kawazumi and Morita.
+Contents
+1.
+Introduction
+1
+2.
+Characteristic classes
+3
+3.
+Twisted cohomology of diffeomorphism groups
+9
+4.
+Cohomology of Torelli groups
+19
+5.
+The case 2n = 2
+22
+References
+28
+1. Introduction
+This paper can be considered as a (somewhat extensive) addendum to our earlier
+work with Kupers [KRW20b]. We shall be concerned with the manifold Wg :=
+#gSn × Sn generalising to higher dimensions the orientable surface of genus g, its
+topological group Diff+(Wg) of orientation-preserving diffeomorphisms, and various
+subgroups of it. The first kind of subgroups are Diff(Wg, D2n) ≤ Diff+(Wg, ∗) ≤
+Diff+(Wg), the diffeomorphisms which fix a disc and a point respectively.
+The
+second kind are their Torelli subgroups
+Tor(Wg, D2n),
+Tor+(Wg, ∗),
+Tor+(Wg),
+consisting of those diffeomorphisms which in addition act trivially on Hn(Wg; Z).
+The intersection form on this middle cohomology group is nondegenerate and (−1)n-
+symmetric, giving a homomorphism
+αg : Diff+(Wg) −→ Gg :=
+�
+Sp2g(Z)
+if n is odd,
+Og,g(Z)
+if n is even.
+This map is not always surjective, but its image is a certain finite-index subgroup
+G′
+g ≤ Gg, even when restricted to Diff(Wg, D2n), so there is an outer G′
+g-action on
+each of the Torelli subgroups. This makes the rational cohomology of each of the
+Torelli groups into G′
+g-representations.
+In [KRW20b], for 2n ≥ 6 we determined H∗(BTor(Wg, D2n); Q) as a ring and
+as a G′
+g-representation in a range of degrees tending to infinity with g.
+Using
+the Serre spectral sequence associated to various simple fibrations relating the dif-
+ferent Torelli groups we were able to also determine H∗(BTor+(Wg, ∗); Q) and
+H∗(BTor+(Wg); Q) as G′
+g-representations. This kind of argument was not able to
+2010 Mathematics Subject Classification. 55R40, 11F75, 57S05, 18D10, 20G05.
+Key words and phrases. Cohomology of diffeomorphism groups, Torelli groups, cohomology of
+arithmetic groups, Miller-Morita-Mumford classes.
+1
+
+2
+OSCAR RANDAL-WILLIAMS
+determine the ring structure, however, as multiplicative information gets lost when
+passing to the associated graded of the Serre filtration. Here we shall determine
+H∗(BTor+(Wg, ∗); Q) and H∗(BTor+(Wg); Q) as Q-algebras too: this is achieved in
+Theorem 4.1. The statement given there is more powerful, but just as in [KRW20b,
+Section 5] one can extract from it the following presentation for H∗(BTor+(Wg); Q),
+which is easier to parse. (A presentation for H∗(BTor+(Wg, ∗); Q) can be extracted
+in a similar way.)
+Let us write H(g) := Hn(Wg; Q), on which G′
+g operates in the evident way. Let
+λ : H(g) ⊗ H(g) → Q denote the intersection form, and {ai}2g
+i=1 be a basis of H(g)
+with dual basis {a#
+i }2g
+i=1 in the sense that λ(a#
+i , aj) = δij, so that the form dual
+to the pairing λ is ω = �2g
+i=1 ai ⊗ a#
+i . In Section 2.2 we will construct certain
+“modified twisted Miller–Morita–Mumford classes”, which when restricted to the
+Torelli group yield G′
+g-equivariant maps
+¯κc : H(g)⊗r −→ Hn(r−2)+|c|(BTor+(Wg); Q)
+for each c ∈ Q[e, p1, p2, . . . , pn−1] = H∗(BSO(2n); Q) and each s ≥ 0. When r = 0
+we write ¯κc = ¯κc(1); these agree with the usual Miller–Morita–Mumford classes κc.
+Theorem A. If 2n ≥ 6 then, in a range of degrees tending to infinity with g,
+H∗(BTor+(Wg); Q) is generated as a Q-algebra by the classes ¯κc(v1 ⊗ · · · ⊗ vr) for
+c a monomial in e, p1, . . . , pn−1, and r ≥ 0, such that n(r −2)+|c| > 0. A complete
+set of relations in this range is given by
+(i) ¯κc(vσ(1) ⊗ · · · ⊗ vσ(r)) = sign(σ)n · ¯κc(v1 ⊗ · · · ⊗ vr),
+(ii) ¯κe(v1) = 0,
+(iii)
+�
+i
+¯κx(v ⊗ ai) · ¯κy(a#
+i ⊗ w) = ¯κx·y(v ⊗ w) +
+1
+χ2 ¯κe2 · ¯κx(v) · ¯κy(w)
+− 1
+χ
+�
+¯κe·x(v) · ¯κy(w) + ¯κx(v) · ¯κe·y(w)
+�
+,
+(iv)
+�
+i
+¯κx(v ⊗ ai ⊗ a#
+i ) = χ−2
+χ ¯κe·x(v) +
+1
+χ2 ¯κe2 · ¯κx(v),
+(v) ¯κLi = 0,
+for v ∈ H(g)⊗r and w ∈ H(g)⊗s.
+For 2n = 4 or 2n = 2 there is still a map from the Q-algebra given by this
+presentation to H∗(BTor+(Wg); Q). If 2n = 2 then (in a stable range) this map
+is an isomorphism onto the maximal algebraic subrepresentation in degrees ≤ N,
+assuming that H∗(BTor+(Wg); Q) is finite-dimensional in degrees < N for all large
+enough g. This is known to hold for N = 2 by work of Johnson [Joh85].
+1.1. Outline. The overall strategy is parallel to [KRW20b]. There we defined cer-
+tain twisted Miller–Morita–Mumford classes and used them to describe the twisted
+cohomology groups H∗(BDiff+(Wg, D2n); H⊗s) in a stable range of degrees, where
+H is the local coefficient system corresponding to Hn(Wg; Q) with the action by dif-
+feomorphisms of Wg. This calculation was valid for 2n = 2 as well. Using that for
+2n ≥ 6 the G′
+g-representations H∗(BTor+(Wg, D2n); Q) extend to representations
+of the ambient algebraic group (namely Sp2g or Og,g) by [KRW20a]1, the argument
+was completed by establishing the degeneration of the Serre spectral sequence
+Ep,q
+2
+= Hp(G′
+g; Hq(BTor+(Wg, D2n); Q)⊗H⊗s) =⇒ Hp+q(BDiff+(Wg, D2n); H⊗s)
+using work of Borel, and then using a categorical form of Schur–Weyl duality to ex-
+tract the structure of H∗(BTor+(Wg, D2n); Q) from the H∗(BDiff+(Wg, D2n); H⊗s)
+for all s’s and various structure maps between them.
+1In fact we did something more complicated in [KRW20b] because this algebraicity result was
+not known at the time, but please allow some narrative leeway.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+3
+The twisted Miller–Morita–Mumford classes may be defined on BDiff+(Wg, ∗)
+too, but not on BDiff+(Wg).
+Our first task will be to define so-called “modi-
+fied twisted Miller–Morita–Mumford classes” in H∗(BDiff+(Wg); H⊗s) and analyse
+their behaviour: it turns out that their behaviour is significantly more complicated
+than the unmodified version, though still understandable. We will then use them
+to describe the twisted cohomology groups H∗(BDiff+(Wg); H⊗s) in a stable range
+of degrees. This description will be in terms of a certain vector space of graphs
+with vertices labelled by monomials in Euler and Pontrjagin classes, which play the
+role here of the vector spaces of labelled partitions from [KRW20b]. The passage
+from this calculation to H∗(BTor+(Wg); Q) is as above.
+The case of dimension 2n = 2 is somewhat special, in precisely the same way as
+it was in [KRW20b]: the calculation of H∗(BDiff+(Wg); H⊗s) is valid in this case,
+but as the cohomology of BTor+(Wg) is not even known to be finite-dimensional
+in a stable range, we cannot make a conclusion about it. (Instead one can make
+a conclusion about the continuous cohomology of the Torelli group, i.e. the Lie
+algebra cohomology of its Mal’cev Lie algebra: see [KRW21], [FNW21], [Hai20].)
+In addition, in this case our modified twisted Miller–Morita–Mumford classes are
+essentially the same as those that have been defined by Kawazumi and Morita
+[Mor96, KM96, KM01], and the graphical calculus that we employ is similar to
+theirs. In Section 5 we fully explain this connection, and also relate it to work of
+Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].
+To avoid a great deal of repetition we have refrained from spelling out a lot of the
+background that was given in [KRW20b], and from giving in detail arguments that
+are very similar to those given there. As such this paper should not be considered
+as attempting to be self-contained: given that its interest will be to readers of
+[KRW20b] this should not present a problem.
+1.2. Acknowledgements. I am grateful to Alexander Kupers for feedback on an
+earlier draft. I was supported by the ERC under the European Union’s Horizon
+2020 research and innovation programme (grant agreement No. 756444) and by a
+Philip Leverhulme Prize from the Leverhulme Trust.
+2. Characteristic classes
+2.1. Recollection on twisted Miller–Morita–Mumford classes. If π′ : E′ →
+X′ is an oriented smooth W 2n
+g -bundle equipped with a section s : X′ → E′, and
+H denotes the local coefficient system x �→ Hn((π′)−1(x); Q) on X′, then it is
+explained in [KRW20b, Section 3.2] that there is a unique class ε = εs ∈ Hn(E′; H)
+characterised by
+(i) for each x ∈ X′ the element ε|(π′)−1(x) ∈ Hn((π′)−1(x); Q)⊗Hn((π′)−1(x); Q)
+is coevaluation, and
+(ii) s∗ε = 0.
+The proof is as follows. The Serre spectral sequence yields an exact sequence
+0 → Hn(X′; H)
+(π′)∗
+→ Hn(E′; H) → H0(X′; H∨⊗H)
+dn+1
+→ Hn+1(X′; H)
+(π′)∗
+→ Hn+1(E′; H)
+and the section s shows that the right-hand map (π′)∗ is injective, so that the map
+dn+1 is zero, and splits the left-hand map (π′)∗. The class coev ∈ H0(X′; H∨ ⊗ H)
+then gives rise to a unique ε satisfying the given properties.
+We then defined the twisted Miller–Morita–Mumford class
+(2.1)
+κεac = κεac(π′, s) :=
+�
+π′ εa · c(Tπ′E′) ∈ H(a−2)n+|c|(X′; H⊗a).
+
+4
+OSCAR RANDAL-WILLIAMS
+2.2. Modified twisted Miller–Morita–Mumford classes. If π : E → X is
+an oriented smooth W 2n
+g -bundle but is not equipped with a section then, as long
+as χ := χ(Wg) = 2 + (−1)n2g ̸= 0 (i.e. (n, g) ̸= (odd, 1), cf. Remark 2.1), the
+cohomological role of the section can instead be played by the transfer map
+1
+χπ!(e · −) : H∗(E; H) −→ H∗(X; H),
+where e := e(TπE) ∈ H2n(E; Q) denotes the Euler class of the vertical tangent
+bundle. The projection formula
+1
+χπ!(e · π∗(x)) = 1
+χπ!(e) · x = χ
+χx = x
+shows that this map splits π∗. Thus in this situation there is a unique class ¯ε ∈
+Hn(E; H) characterised by
+(i) for each x ∈ X the element ¯ε|π−1(x) ∈ Hn(π−1(x); Q) ⊗ Hn(π−1(x); Q) is
+coevaluation, and
+(ii)
+1
+χπ!(e · ¯ε) = 0.
+Remark 2.1. If (n, g) = (odd, 1) then there is no class ¯ε ∈ Hn(E; H) satisfying (i)
+and natural under pullback. To see this it suffices to give one example of a smooth
+oriented W1-bundle for which ¯ε does not exist. Consider the Borel construction for
+the evident action of S1 × S1 on W1 = Sn × Sn given by considering Sn as the unit
+sphere in C(n+1)/2. This gives a smoth oriented W1-bundle over B(S1 × S1) with
+total space E ≃ CP(n−1)/2 × CP(n−1)/2. Thus Hn(E; H) = 0 as n is odd but E has
+a cell structure with only even-dimensional cells.
+By analogy with (2.1) we may then define the modified twisted Miller–Morita–
+Mumford class
+(2.2)
+κ¯εac = κ¯εac(π) :=
+�
+π
+¯εa · c(TπE) ∈ H(a−2)n+|c|(X; H⊗a).
+If π : E → X does have a section s : X → E then the class ε ∈ Hn(E; H) is also
+defined, and we may compare it with ¯ε as follows:
+Lemma 2.2. If π : E → X has a section then ¯ε = ε − 1
+χπ∗κεe.
+Proof. The classes ε, ¯ε ∈ Hn(E; H) are both defined, and agree when restricted to
+the fibres of the map π, so by considering the Serre spectral sequence for π we must
+have ¯ε − ε = π∗(x) for some class x ∈ Hn(X; H). Applying 1
+χπ!(e · −) we see that
+x = 1
+χπ!(e · (¯ε − ε)) = 0 − 1
+χπ!(e · ε) = − 1
+χκεe. (Here we have used, as we often will,
+the fact that e has even degree to commute it past ε.)
+□
+Remark 2.3 (Splitting principle). The pullback
+(2.3)
+E1 ×X E2
+E2
+E1
+X,
+pr1
+pr2
+π2
+π1
+where πi : Ei → X are copies of the map π, is equipped with a section given by the
+diagonal map ∆ : E1 → E1 ×X E2. As the maps π∗
+1 and pr∗
+2 are injective (they are
+split by their corresponding transfer maps), for the purpose of establishing identities
+between the characteristic classes we have discussed it suffices to do so for bundles
+which do have a section.
+There is another description of ¯ε which is sometimes useful. Let pr1 : E ×X E →
+E be as in (2.3), which is an oriented Wg-bundle with section given by the diagonal
+map ∆, and so has the class κεe(pr1, ∆) defined.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+5
+Lemma 2.4. We have ¯ε = − 1
+χκεe(pr1, ∆) ∈ Hn(E; H).
+Proof. By Remark 2.3 we may suppose without loss of generality that π : E → X
+has a section s : X → E, defining a class ε = εs ∈ Hn(E; H).
+Consider the
+pullback square (2.3); let ei = (pri)∗(e) ∈ H2n(E1 ×X E2; Q) be the Euler class
+of the vertical tangent bundle on the ith factor. Considering pr1 as a Wg-bundle
+with section given by the diagonal map ∆, there is a class ε∆ ∈ Hn(E1 ×X E2; H)
+defined. As both ε∆ and pr∗
+2(εs) restrict to coevaluation on the fibres of pr1, we
+have ε∆ − pr∗
+2(εs) = pr∗
+1(x) for some class x ∈ Hn(E1; H). Pulling this equation
+back along ∆ shows that x = −εs, so ε∆ = pr∗
+2(εs) − pr∗
+1(εs). Then we have
+κεe(pr1, ∆) = (pr1)!(ε∆ · e2)
+= (pr1)!((pr∗
+2(εs) − pr∗
+1(εs)) · e2)
+= (pr1)!(pr∗
+2(εs · e)) − (pr1)!(pr∗
+1(εs) · e2)
+= π∗
+1(π2)!(εs · e) − χεs
+= π∗
+1κεe(π, s) − χεs
+= −χ¯ε
+as required.
+□
+The intersection form of the fibres of π : E → X provides a map of local coeffi-
+cient systems λ : H⊗H → Q; as we will often be concerned with applying it to two
+factors of a tensor power H⊗k and will have to specify which factors we apply it to,
+we will denote λ by λ1,2 and more generally write λi,j : H⊗k → H⊗k−2 for the map
+that applies λ to the ith and jth factors. We call such operations contraction.
+If p : E1 ×X E2 → X denotes the fibre product of two copies of π : E → X, and
+if this has a section s : X → E, then in [KRW20b, Lemma 3.9] we have established
+the formula
+(2.4)
+λ1,2(ε × ε) = ∆!(1) − 1 × v − v × 1 + p∗s∗e ∈ H2n(E1 ×X E2; Q),
+where v = s!(1) ∈ H2n(E; Q) is the fibrewise Poincar´e dual to the section s, cf.
+[KRW20b, Lemma 3.1]. The analogue of this formula for ¯ε is as follows.
+Lemma 2.5. We have
+λ1,2(¯ε × ¯ε) = ∆!(1) +
+1
+χ2 p∗κe2 − 1
+χ(e × 1 + 1 × e) ∈ H2n(E1 ×X E2; Q).
+Proof. As in Remark 2.3 we may suppose without loss of generality that π : E → X
+has a section s : X → E, so that ε ∈ Hn(E; H) is defined.
+By Lemma 2.2 we have ¯ε = ε − 1
+χπ∗κεe ∈ Hn(E; H), and so
+λ1,2(¯ε × ¯ε) = λ1,2((ε − 1
+χπ∗κe·ε) × (ε − 1
+χπ∗κe·ε))
+= λ1,2(ε × ε) − λ1,2( 1
+χπ∗κεe × ε)
+− λ1,2(ε × 1
+χπ∗κεe) + λ1,2( 1
+χπ∗κεe × 1
+χπ∗κεe).
+The first term is given by (2.4), and using [KRW20b, Proposition 3.10] the last
+term is given by
+λ1,2( 1
+χπ∗κεe × 1
+χπ∗κεe) =
+1
+χ2 p∗λ1,2(κεe · κεe) =
+1
+χ2 p∗(κe2 + (χ2 − 2χ)s∗e).
+For the middle two terms, note that
+ε × 1
+χπ∗κεe = 1
+χ(ε × 1) · p∗(κe·ε) = 1
+χ(ε · π∗κεe) × 1
+
+6
+OSCAR RANDAL-WILLIAMS
+so we need to calculate λ1,2(ε · π∗κεe) ∈ H2n(E; Q). The class ε · κεe is the fibre
+integral along pr1 : E1 ×X E2 → E1 of ε × (ε · e) = (ε × ε) · (1 × e), so
+λ1,2(ε · κεe) = (pr1)!(λ1,2(ε × ε) · (1 × e))
+= (pr1)!((∆!(1) − 1 × v − v × 1 + p∗s∗e) · (1 × e))
+= e − π∗s∗e − χv + χπ∗s∗e
+and hence
+λ1,2(ε × 1
+χκεe) = 1
+χ(e − π∗s∗e − χv + χπ∗s∗e) × 1
+= 1
+χe × 1 + χ−1
+χ p∗s∗e − v × 1
+and similarly
+λ1,2( 1
+χκεe × ε) = 1
+χ1 × e + χ−1
+χ p∗s∗e − 1 × v.
+Combining these gives
+λ1,2(¯ε × ¯ε) = ∆!(1) − 1 × v − v × 1 + p∗s∗e
++
+1
+χ2 p∗κe2 + χ−2
+χ p∗s∗e
+− ( 1
+χe × 1 + χ−1
+χ p∗s∗e − v × 1)
+− ( 1
+χ1 × e + χ−1
+χ p∗s∗e − 1 × v)
+= ∆!(1) +
+1
+χ2 p∗κe2 − 1
+χ(e × 1 + 1 × e)
+as required.
+□
+If in addition we have a lift ℓ : E → B of the fibrewise Gauss map along some
+fibration θ : B → BSO(2n) then for any c ∈ H∗(B; Q) we can define modified
+twisted Miller–Morita–Mumford classes by the formula
+κ¯εac := π!(¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X; H⊗a).
+Under the action of a permutation σ ∈ Sa of the tensor factors these classes
+transform as sign(σ)n, as ¯ε has degree n.
+Thus for any finite set S there is a
+well-defined element
+(2.5)
+κ¯εSc := π!(¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X; H⊗S) ⊗ (det QS)⊗n.
+To keep track of signs, for an ordered set S = {s1 < s2 < · · · < sa} we will often
+write κ¯εs1,...,sac ∈ Hn(a−2)+|c|(X; H⊗S) for the corresponding element, understand-
+ing that if σ is a reordering of S then κ¯εσ(s1),...,σ(sa)c = sign(σ)nκ¯εs1,...,sa c.
+Using Lemma 2.5 we immediately see that these characteristic classes satisfy the
+following analogue of the contraction formula from [KRW20b, Proposition 3.10].
+Proposition 2.6 (Modified contraction formula). In H∗(X; H⊗−) we have the
+identities
+λ1,2(π!(¯ε1,2,...,a · ℓ∗c)) = ( χ−2
+χ )π!(¯ε3,4,...,a · ℓ∗(e · c)) +
+1
+χ2 κe2 · π!(¯ε3,4,...,a · ℓ∗c)
+and
+λa,a+1(π!(¯ε1,2,...,a · ℓ∗c) · π!(¯εa+1,...,a+b · ℓ∗c′)) = π!(¯ε1,...,a−1,a+2,...,a+b · ℓ∗(c · c′))
++
+1
+χ2 κe2 · π!(¯ε1,...,a−1 · ℓ∗c) · π!(¯εa+2,...,a+b · ℓ∗c′)
+− 1
+χπ!(¯ε1,...,a−1 · ℓ∗(e · c)) · π!(¯εa+2,...,a+b · ℓ∗c′)
+− 1
+χπ!(¯ε1,...,a−1 · ℓ∗c) · π!(¯εa+2,...,a+b · ℓ∗(e · c′)).
+Similarly, from Lemma 2.2 we immediately obtain the following:
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+7
+Proposition 2.7. If the bundle π : E → X has a section, so that the class ε and
+hence κεSc is defined, then
+κ¯εSc =
+�
+I⊆S
+κεIc(− 1
+χκεe)S\I ∈ H∗(X; H⊗S) ⊗ (det QS)⊗n.
+Let us give an example of using the modified contraction formula to evaluate an
+expression.
+Example 2.8. Consider the class λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6). Then
+λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = λ1,5λ2,6
+�
+κ¯ε1,2,5,6 +
+1
+χ2 κe2κ¯ε1,2κ¯ε5,6
+− 1
+χ(κ¯ε1,2eκ¯ε5,6 + κ¯ε1,2κ¯ε5,6e)
+�
+.
+The first term is
+λ1,5λ2,6(κ¯ε1,2,5,6) = (−1)nλ1,5λ2,6(κ¯ε1,5,2,6)
+= (−1)nλ2,6( χ−2
+χ κ¯ε2,6e +
+1
+χ2 κe2κ¯ε2,6)
+= (−1)n χ−2
+χ ( χ−2
+χ κe2 +
+1
+χ2 κe2χ) + (−1)n 1
+χ2 κe2( χ−2
+χ χ)
+= (−1)n( (χ−2)2
+χ2
++ 2 χ−2
+χ2 )κe2.
+The second term is
+1
+χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε5,6) = (−1)n 1
+χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε6,5)
+= (−1)n 1
+χ2 κe2λ1,5(κ¯ε1,5)
+= (−1)n χ−2
+χ2 κe2.
+The third term is
+− 1
+χλ1,5λ2,6(κ¯ε1,2eκ¯ε5,6) = (−1)n+1 1
+χλ1,5λ2,6(κ¯ε1,2eκ¯ε6,5)
+= (−1)n+1 1
+χλ1,5(κ¯ε1,5e − 1
+χκ¯ε1eκ¯ε5e)
+= (−1)n+1 1
+χ
+�
+( χ−2
+χ κe2 +
+1
+χ2 κe2χ)
+− 1
+χ(κe2 +
+1
+χ2 κe2χ2 − 1
+χ(2χκe2))
+�
+= (−1)n+1( χ−2
+χ2 +
+1
+χ2 −
+1
+χ2 −
+1
+χ2 +
+2
+χ2 )κe2
+= (−1)n+1 χ−1
+χ2 κe2
+and the fourth term is the same as the third by the evident symmetry.
+In total we have
+λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = (−1)n( (χ−2)2
+χ2
++ 2 χ−2
+χ2 + χ−2
+χ2 − 2 χ−1
+χ2 )κe2
+= (−1)n χ−3
+χ κe2.
+□
+2.3. Graphical interpretation. In [KRW20b, Section 5] it was found to be very
+convenient to adopt a graphical formalism where κεac corresponds to a vertex with
+a half-edges incident to it and a formal label c, a product of κεac’s corresponds to
+a disjoint union of such vertices, and applying the contraction λi,j corresponds to
+pairing up the half-edges labelled i and j.
+It will be convenient to adopt a similar formalism here. Let S be a finite set, and
+V be a graded Q-algebra with a distinguished element e ∈ V2n. Slightly modifying2
+2The difference is that we allow labelled vertices whose contribution to the degree is 0.
+
+8
+OSCAR RANDAL-WILLIAMS
+the definition from [KRW20b, Proof of Theorem 5.1], a marked oriented graph with
+legs S and labelled by V consists of the following data:
+(i) a totally ordered finite set ⃗V (of vertices), a totally ordered finite set ⃗H (of
+half-edges), and a monotone function a: ⃗H → ⃗V (encoding that a half-edge
+h is incident to the vertex a(h)),
+(ii) an ordered matching m = {(ai, bi)}i∈I of the set H ⊔ S (encoding the
+oriented edges of the graph),
+(iii) a function c: V → V with homogeneous values, such that |c(v)|+n(|a−1(v)|−
+2) ≥ 0.
+Marked oriented graphs Γ = (⃗V , ⃗H, a, m, c) and Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) with the
+same set of legs S are isomorphic if there are order-preserving bijections ⃗V
+∼
+→ ⃗V ′
+and ⃗H
+∼
+→ ⃗H′ which intertwine a and a′, intertwine c and c′, and send m to m′. An
+oriented graph is an isomorphism class [Γ] of marked oriented graph. We assign to
+a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) the degree
+deg(Γ) :=
+�
+v∈V
+�
+|c(v)| + n(|a−1(v)| − 2)
+�
+= n(|H| − 2|V |) +
+�
+v∈V
+|c(v)|.
+Let π : E → X be an oriented Wg-bundle with a lift ℓ : E → B of the map clas-
+sifying the vertical tangent bundle along θ : B → BSO(2n), and let V := H∗(B; Q)
+and e := θ∗e ∈ V2n. Then given a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) with
+legs S we form a class
+¯κ(Γ) ∈ Hdeg(Γ)(X; H⊗S)
+by the following recipe. Firstly, we may form
+(2.6)
+�
+v∈V
+κ¯εa−1(v)c(v) ∈ H∗(X; H⊗H),
+where we have used the ordering on ⃗V to order the product, and the ordering on
+⃗H to trivialise the factor of (det QH)⊗n = (�
+v∈V det Qa−1(v))⊗n that arises from
+(2.5). Secondly, taking two copies S1 and S2 of the set S and writing si ∈ Si for
+the element corresponding to s ∈ S we can form
+(2.7)
+�
+s∈S
+κ¯εs1,s2 ∈ H∗(X, H⊗(S1⊔S2)).
+As each κ¯εs1,s2 has degree 0, the product does not depend on how the factors are
+ordered. Taking the product of (2.6) and (2.7) and then applying λx,y for each
+ordered pair (x, y) in the matching m on H ⊔ S = H ⊔ S1 gives the required class
+¯κ(Γ) ∈ H∗(X; H⊗S2) = H∗(X; H⊗S).
+Example 2.9. In this graphical interpretation we recognise the class evaluated
+in Example 2.8 as that associated to the theta-graph with a certain ordering and
+orientation.
+Clearly ¯κ(Γ) only depends on the underlying oriented graph [Γ]. We now describe
+how it transforms when the orderings on V , H, and the pairs m are changed, without
+changing the underlying labelled graph. If Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) is another marked
+oriented graph and there are bijections f : H → H′ and g : V → V ′ intertwining
+a and a′ and c and c′ and such that under these bijections the matching m′ differs
+from m by reversing the order of k pairs, then
+¯κ(Γ′) = (−1)nksign(f)sign(g) · ¯κ(Γ)
+for certain signs described on [KRW20b, pp. 55-56].
+Graphs considered as representing ¯κ’s behave differently to those representing κ’s
+described in [KRW20b, Section 5]. To distinguish them we will depict the graphs
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+9
+representing κ’s in red, as we did in that paper, and the graphs representing ¯κ’s in
+blue. The contraction formula of [KRW20b, Proposition 3.10] was interpreted in
+[KRW20b, Section 5] as giving relations among red graphs which yield equivalent
+κ-classes. In the generality of a smooth oriented Wg-bundle π : E → X with section
+s : X → E these may be depicted as follows:
+•c
+=
+•ce
++s∗e
+•c
+−2s∗c
+•c
+•c′
+=
+•cc′
++s∗e
+•c
+•c′
+−s∗c
+•c′
+−s∗c′
+•c
+Figure 1. The contraction formula, displayed graphically.
+Here the negative terms only arise when they make sense, i.e. when the vertex
+has valence 2 in the first case, when the vertex labelled c has valence 1 in the second
+case, and when the vertex labelled c′ has valence 1 in the third case.
+Convention. In these and the following figures, to avoid clutter we have adopted the
+following ordering conventions: vertices are numbered starting from 1 from left to
+right, half-edges around each vertex are ordered clockwise starting from the marked
+half-edge, and edges are oriented from the smaller half-edge to the larger one.
+Similarly, the modified contraction formula of Proposition 2.6 can be interpreted
+as giving the following relations among blue graphs which yield equivalent ¯κ-classes:
+•c
+=
+χ−2
+χ
+•ce
++ 1
+χ2
+•c
+•e2
+•c
+•c′
+=
+•cc′
++ 1
+χ2
+•c
+•c′
+•e2
+− 1
+χ(
+•ce
+•c′
++
+•c
+•
+c′e
+)
+Figure 2. The modified contraction formula, displayed graphically.
+3. Twisted cohomology of diffeomorphism groups
+The main goal of this section is to describe the twisted cohomology groups
+H∗(BDiff+(Wg); H⊗S) and H∗(BDiff+(Wg, ∗); H⊗S)
+in a stable range of degrees, of the classifying space BDiff+(Wg) of the group of
+orientation-preserving diffeomorphisms of Wg (which classifies oriented Wg-bundles),
+and the classifying space BDiff+(Wg, ∗) of the group of orientation-preserving dif-
+feomorphisms of Wg which fix a point ∗ ∈ Wg (which classifies oriented Wg-bundles
+with section). In [KRW20b, Theorem 3.15] the analogous calculation was given for
+
+10
+OSCAR RANDAL-WILLIAMS
+the classifying space BDiff(Wg, D2n) of the group of diffeomorphisms of Wg which
+fix a disc D2n ⊂ Wg.
+In order to do this we will also discuss the manifolds Wg equipped with θ-
+structures for the tangential structure θ : BSO(2n)⟨n⟩ → BO(2n), i.e. the n-
+connected cover of BO(2n). In this case we will consider the homotopy quotients
+BDiffθ(Wg) := Bun(T Wg, θ∗γ2n)//Diff(Wg)
+BDiffθ(Wg, ∗) := Bun(T Wg, θ∗γ2n)//Diff(Wg, ∗)
+where Bun(T Wg, θ∗γ2n) denotes the space of vector bundle maps T Wg → θ∗γ2n
+from the tangent bundle of Wg to the bundle classified by θ. The group Diff(Wg)
+acts on the space of bundle maps by precomposing with the derivative.
+There is a factorisation θ : BSO(2n)⟨n⟩
+θor
+→ BSO(2n)
+σ→ BO(2n), and by ob-
+struction theory one sees that the space Bun(T Wg, σ∗γ2n) has two contractible
+path components corresponding to the two orientations of Wg. In particular there
+are equivalences
+Bun(T Wg, σ∗γ2n)//Diff(Wg) ≃ BDiff+(Wg)
+Bun(T Wg, σ∗γ2n)//Diff(Wg, ∗) ≃ BDiff+(Wg, ∗)
+and so θor induces maps
+BDiffθ(Wg) −→ BDiff+(Wg)
+and
+BDiffθ(Wg, ∗) −→ BDiff+(Wg, ∗).
+It is shown in [GRW19, Section 5.2] that these are principal SO[0, n − 1]-fibrations.
+In particular the spaces BDiffθ(Wg) and BDiffθ(Wg, ∗) are path-connected.
+3.1. Spaces of graphs. Our description of the twisted cohomology groups of
+BDiff+(Wg), BDiff+(Wg, ∗), BDiff(Wg, D2n), BDiffθ(Wg) and BDiffθ(Wg, ∗) in a
+stable range will be—via the graphical interpretation given in Section 2.3—in terms
+of graded vector spaces of labelled graphs, modulo certain relations. (Readers of
+[KRW20b] may have been expecting vector spaces of labelled partitions instead:
+here we have found spaces of graphs more convenient for formulating results, cf.
+Remark 3.2, though spaces of labelled partitions will still play a role in the proofs.)
+To describe these spaces of graphs we will use the graded Q-algebras
+V := H∗(BSO(2n)⟨n⟩; Q) = Q[p⌈ n+1
+4
+⌉, . . . , pn−1, e]
+W := H∗(BSO(2n); Q) = Q[p1, . . . , pn−1, e]
+with distinguished elements e of degree 2n given by the Euler class. In order to work
+in a way which is agnostic about the genus g of the manifold Wg under consideration,
+we will work over the ring Q[χ±1] instead of Q, where χ is an invertible formal
+parameter which will—later—be set to the Euler characteristic 2 + (−1)n2g of Wg.
+Definition 3.1.
+(i) Let
+Graph1(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼
+where ∼
+(a) imposes the sign rule for changing orderings of vertices and half-edges and
+for reversing orientations of edges;
+(b) imposes linearity in the labels, and sets a graph containing an a-valent
+vertex labelled by c with |c| + n(a − 1) < 0 to zero;
+(c) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, and if 2n ≡ 0
+mod 4 sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0;
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+11
+(d) imposes the contraction relations3
+•c
+=
+•ce
+−
+2c
+•c
+•c′
+=
+•cc′
+−
+c
+•c′
+−
+c′
+•c
+where the negative terms only arises when they makes sense, i.e. in the
+first case when the vertex has valence 2 and its label c is a scalar multiple
+of 1 ∈ V0, in the second case when c is a scalar multiple of 1 ∈ V0 and has
+valence 1, and similarly in the third case.
+(ii) Let
+Graphθ
+∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V} ⊗ V/ ∼
+where ∼ imposes (a)–(c) as well as
+(d′) imposes the contraction relations of Figure 1.
+(iii) Let
+Graph∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W} ⊗ W/ ∼
+where ∼ imposes (a) and (b), as well as
+(c′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent
+vertex labelled by any degree 2n monomial in Pontrjagin classes equal
+to 0, and for any 1 ≤ i ≤ ⌊n/4⌋ sets
+•
+cpi
+= 1
+χ
+•c
+⊗pi
+and (d′).
+(iv) Let
+Graphθ(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼
+where ∼ imposes (a) and (b), as well as
+(c′′′) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, if 2n ≡ 0 mod 4
+sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0, and sets the
+1-valent vertex labelled by e ∈ V2n equal to 0,
+(d′′′) imposes the contraction relations of Figure 2.
+(v) Let
+Graph(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W}/ ∼
+where ∼ imposes (a), (b), as well as
+(c′′′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent
+vertex labelled by any degree 2n monomial in Pontrjagin classes equal
+to 0, sets the 1-valent vertex labelled by e ∈ W2n equal to 0, and for
+any 1 ≤ i ≤ ⌊n/4⌋ sets
+3These are the relations from Figure 1 when s∗ kills all positive-degree classes.
+
+12
+OSCAR RANDAL-WILLIAMS
+•
+cpi
+= 1
+χ
+•c
+•
+epi
+and (d′′′).
+Remark 3.2 (Graphs and partitions). In all cases one can apply the (modified)
+contraction formula to pass from a graph to a sum of graphs with strictly fewer
+edges, and so by iterating to a sum of graphs with no edges. These are disjoint
+unions of labelled corollas, and so correspond to partitions of S with labels in V or
+W, plus additional external labels in cases (ii) and (iii). There are two issues with
+this. The first is that in cases (iv) and (v) it is not clear that the resulting sum of
+disjoint unions of labelled corollas is unique, as one has to choose an order in which
+to eliminate edges. The second is that even if it is, then the functoriality on the
+Brauer category which we describe below would involve gluing in edges and then
+eliminating them, leading to a complicated formula. This is why we have found it
+convenient to work with spaces of graphs.
+We wish to consider each of the above as defining functors on the (signed) Brauer
+category as in [KRW20b, Section 2.3], but to take into account the parameter χ we
+must slightly generalise to a Q[χ]-linear version of the (signed) Brauer category.
+Definition 3.3. For finite sets S and T let preBrχ(S, T ) be the free Q[χ]-module
+on tuples (f, mS, mT ) of a bijection f from a subset S◦ ⊂ S to a subset T ◦ ⊂ T ,
+an ordered matching mS of S \ S◦, and an ordered matching mT of T \ T ◦.
+Let Brχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) −
+(f, m′
+S, m′
+T ) whenever mS agrees with m′
+S after reversing some pairs, and mT agrees
+with m′
+T after reversing some pairs.
+Let sBrχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) −
+(−1)kl(f, m′
+S, m′
+T ) whenever mS agrees with m′
+S after reversing k pairs, and mT
+agrees with m′
+T after reversing l pairs.
+Let (s)Brχ be the Q[χ]-linear category whose objects are finite sets, and whose
+morphisms are the Q[χ]-modules (s)Brχ(S, T ) defined above. In the case of Brχ
+we think of [f, mS, mT ] as representing 1-dimensional cobordisms with no closed
+components: then the composition law is given by composing cobordisms and then
+replacing each closed 1-manifold by a factor of χ − 2. In the case of sBrχ we think
+of (f, mS, mT ) as representing oriented 1-dimensional cobordisms with no closed
+components: then the composition law is given by composing cobordisms and then
+replacing each compatibly oriented closed 1-manifold by a factor of −(χ − 2).
+Let d(s)Brχ denote the subcategories having all objects and morphisms spanned
+by [f, mS, mT ] with T ◦ = ∅.
+For a central charge d ∈ Q let (d)(s)Brd denote the Q-linear category obtained
+by specialising the Q[χ]-linear category (d)(s)Brχ to χ = 2+d for (d)Br or χ = 2−d
+for (d)sBr. (This notation then agrees with [KRW20b, Definition 2.14, 2.19].)
+We consider the spaces of graphs above as defining Q[χ]-linear functors
+Graph1(−), Graphθ
+∗(−), Graph∗(−), Graphθ(−), Graph(−) : (s)Brχ → Gr(Q[χ±1]-mod)
+in the evident way, by gluing of oriented graphs (after orientations have been ar-
+ranged to be compatible). We endow them with a lax symmetric monoidality by
+disjoint union of graphs. We write Graph1(−)g : (s)Br2g → Gr(Q-mod) and so on
+for their specialisations at χ = 2 + (−1)n2g (defined for (n, g) ̸= (odd, 1)).
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+13
+3.2. The isomorphism theorem. Theorem 3.4 below extends [KRW20b, Theo-
+rem 3.15] to BDiffθ(Wg, ∗), BDiff+(Wg, ∗), BDiffθ(Wg), and BDiff+(Wg).
+To formulate it we first observe that when π : E → X is a smooth oriented
+Wg-bundle and H is the local coefficient system over X given by the fibrewise nth
+homology of this bundle, the fibrewise intersection form λ : H⊗H → Q and its dual
+ω : Q → H ⊗ H are (−1)n-symmetric and satisfy λ ◦ ω = (−1)n2g · Id, so provide a
+Q-linear functor S �→ H⊗S from (s)Br2g to the category of local coefficient systems
+of Q-modules over X. (Strictly speaking our definitions require χ = 2 + (−1)n2g to
+be invertible, so we omit the case (n, g) = (odd, 1).) Composing this with taking
+cohomology gives a functor
+H∗(X; H⊗−) : (s)Br2g −→ Gr(Q-mod)
+S �−→ H∗(X; H⊗S).
+The relations in the various spaces of graphs defined in Section 3.1 were chosen
+precisely to match the contraction formula of [KRW20b, Proposition 3.10] (in the
+case of Graph1) and the modified contraction formula of Proposition 2.6 (in the
+other cases), so that assigning to a graph its associated κ- or ¯κ-class provides
+natural transformations
+(i) κ : Graph1(−)g → H∗(BDiff(Wg, D2n); H⊗−),
+(ii) κ : Graphθ
+∗(−)g → H∗(BDiffθ(Wg, ∗); H⊗−),
+(iii) κ : Graph∗(−)g → H∗(BDiff+(Wg, ∗); H⊗−),
+(iv) ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg); H⊗−),
+(v) ¯κ : Graph(−)g → H∗(BDiff+(Wg); H⊗−),
+of functors (s)Br2g → Gr(Q-mod).
+Theorem 3.4. For 2n = 2 or 2n ≥ 6 the maps (i)–(v) are isomorphisms in a
+range of cohomological degrees tending to infinity with g.
+We will first give the proof in cases (i), (ii), (iii), and in case (v) assuming case
+(iv); the much more involved case (iv) will be treated afterwards.
+Proof of Theorem 3.4 (i), (ii), (iii), (v). For case (i) observe that Graph1(−)g is
+naturally isomorphic to the functor G(−, V) from [KRW20b, Proof of Theorem 5.1],
+which is shown there to be isomorphic to the functor P(−, V)≥0 ⊗ det⊗n. This case
+then follows from [KRW20b, Theorem 3.15].
+For case (ii) we first construct the homotopy fibre sequence
+(3.1)
+BDiff(Wg, D2n) −→ BDiffθ(Wg, ∗) −→ BSO(2n)⟨n⟩.
+The left-hand term may be written as the homotopy quotient of Diff(Wg, ∗) acting
+on the Stiefel manifold Fr(T∗Wg) given by the space of frames in the tangent space to
+Wg at the point ∗ ∈ Wg, as this action is transitive and its stabiliser is the subgroup
+which fixes a point and its tangent space, which is homotopy equivalent to fixing a
+disc. The middle term was defined as the homotopy quotient of Diff(Wg, ∗) acting
+on Bun(T Wg, θ∗γ2n). Evaluation at ∗ ∈ Wg defines a Diff(Wg, ∗)-invariant map
+ev : Bun(T Wg, θ∗γ2n) −→ BSO(2n)⟨n⟩,
+which is a fibration.
+If we choose a point x ∈ BSO(2n)⟨n⟩ and a framing ξ :
+(θ∗γ2n)x
+∼
+→ R2n, then there is a map ξ∗ : ev−1(x) → Fr(T∗Wg) given by sending a
+bundle map ˆℓ : T Wg → θ∗γ2n whose underlying map sends ∗ to x to the framing
+ξ ◦ ˆℓx : T∗Wg → (θ∗γ2n)x → R2n. One verifies by obstruction theory that ξ∗ :
+ev−1(x) → Fr(T∗Wg) is a weak equivalence. Taking homotopy orbits for Diff(Wg, ∗)
+then gives the required homotopy fibre sequence.
+
+14
+OSCAR RANDAL-WILLIAMS
+As H∗(BDiff(Wg, D2n); H⊗S) is spanned by products of twisted Miller–Morita–
+Mumford classes κεac with c ∈ V in a stable range by (i), and these classes may be
+defined on BDiffθ(Wg, ∗), the Serre spectral sequence
+H∗(BSO(2n)⟨n⟩; Q) ⊗ H∗(BDiff(Wg, D2n); H⊗S) ⇒ H∗(BDiffθ(Wg, ∗); H⊗S)
+for the homotopy fibre sequence (3.1) collapses in a stable range. The result then
+follows by observing that the analogue of the Serre filtration of Graphθ
+∗(−)g, induced
+by the descending filtration by degrees of H∗(BSO(2n)⟨n⟩; Q) = V, has
+gr(Graphθ
+∗(−)g) ∼= V ⊗ Graph1(−)g,
+because modulo V>0 the formula of (d′) specialises to that of (d). The induced map
+gr(κ) : gr(Graphθ
+∗(−)g) −→ gr(H∗(BDiffθ(Wg, ∗); H⊗−))
+therefore has the form V ⊗ {the map κ in case (i)} so is an isomorphism in a stable
+range by case (i). Case (ii) follows.
+Case (iii) is just like the above, using the homotopy fibre sequence
+BDiff(Wg, D2n) −→ BDiff+(Wg, ∗) −→ BSO(2n)
+instead, which is established in the analogous way, and W in place of V.
+Case (v) can be deduced from case (iv) by applying the same method to the
+homotopy fibre sequence
+BDiffθ(Wg) −→ BDiff+(Wg)
+ξ
+−→ BSO[0, n]
+established in [GRW19, Section 5.2]. The filtration step is a little different, so we
+give some details. It follows from (iv) that H∗(BDiffθ(Wg); H⊗S) is spanned by
+products of twisted Miller–Morita–Mumford classes κ¯εac with c ∈ V in a stable
+range, and these may be defined on BDiff+(Wg) (in fact they may be defined even
+for c ∈ W) so the corresponding Serre spectral sequence
+H∗(BSO[0, n]; Q) ⊗ H∗(BDiffθ(Wg); H⊗S) ⇒ H∗(BDiff+(Wg); H⊗S)
+degenerates in a stable range. In this case the analogue of the Serre filtration on
+Graph(−)g is induced by giving the graph Υi := ({0}, ∅, ∅ → {0}, ∅, c(0) = epi)
+filtration 4i for 1 ≤ i ≤ ⌊n/4⌋, giving all other connected graphs filtration 0, and
+extending multiplicatively. The associated graded of this filtration has the form
+gr(Graph(−)g) ∼= Q[Υ1, Υ2, . . . , Υ⌊n/4⌋] ⊗ Graphθ(−)g,
+because the relation in (c′′′′) shows that any graph with a vertex labelled cpi for
+1 ≤ i ≤ ⌊n/4⌋ is equivalent to a graph of strictly larger filtration, unless the vertex
+is 0-valent and the label is epi. As ¯κ(Υi) = κepi = χ · ξ∗(pi) by [GRW19, Remark
+5.5] it follows that the induced map
+gr(¯κ) : gr(Graph(−)g) −→ gr(H∗(BDiff+(Wg); H⊗S))
+has the form {an isomorphism} ⊗ {the map ¯κ in case (iv)} so is an isomorphism in
+a stable range by case (iv).
+□
+3.3. Proof of Theorem 3.4 (iv). The proof of Theorem 3.4 (iv) is of a less formal
+nature. It will be parallel to that of [KRW20b, Theorem 3.15], but algebraically
+more complicated.
+An important tool will be the following lemma, inspired by
+[Qui71, p. 566].
+Lemma 3.5. Let G be a topological group and p : P → X be a principal G-bundle
+with action a : G×P → P, which satisfies the Leray–Hirsch property in cohomology
+over a field F. Then
+H∗(X; F)
+H∗(P; F)
+H∗(G; F) ⊗F H∗(P; F)
+p∗
+a∗
+1⊗Id
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+15
+is an equaliser diagram.
+Proof. Let us leave F implicit.
+By the Leray–Hirsch property H∗(P) is a free
+H∗(X)-module and hence is faithfully flat. Thus it suffices to prove that the dia-
+gram is an equaliser diagram after applying −⊗H∗(X) H∗(P). By the Leray–Hirsch
+property we also have H∗(P) ⊗H∗(X) H∗(P)
+∼
+→ H∗(P ×X P). Thus it suffices to
+show that
+H∗(P)
+H∗(P ×X P)
+H∗(G) ⊗ H∗(P ×X P)
+pr∗
+2
+a∗
+1⊗Id
+is an equaliser diagram, which is the same question for the principal G-bundle
+pr2 : P ×X P → P. But this principal G-bundle has a section given by the diagonal
+map, which trivialises it: this trivialisation identifies the diagram with
+H∗(P)
+H∗(G) ⊗ H∗(P)
+H∗(G) ⊗ H∗(G) ⊗ H∗(P)
+1⊗Id
+µ∗⊗Id
+1⊗Id
+which is indeed an equaliser diagram as it has a contraction induced by a∗.
+□
+We adapt the proof of [KRW20b, Theorem 3.15], supposing for concreteness that
+n is odd. Consider the tangential structure θ × Y : BSO(2n)⟨n⟩ × Y → BSO(2n)
+with Y = K(W ∨, n + 1) and W a generic rational vector space. Then we have
+H∗(Y ; Q) ∼= Sym∗(W[n + 1]), the symmetric algebra on the vector space W places
+in (even) degree n + 1. If n is even then like at the end of the proof of [KRW20b,
+Theorem 3.15] we would take Y = K(W ∨, n + 2) instead, so H∗(Y ; Q) would still
+be a symmetric algebra. Apart from this there is no essential difference, and we
+will not comment further on the differences in the case n even.
+There are associated universal Wg-bundles
+π : Eθ −→ BDiffθ(Wg)
+πY : Eθ×Y −→ BDiffθ×Y (Wg)
+and an evaluation map ℓ : Eθ×Y → Y . Neglecting the “maps to Y ” part of the
+tangential structure gives a homotopy fibre sequence
+map(Wg, Y ) −→ BDiffθ×Y (Wg) −→ BDiffθ(Wg).
+We can take Y to be a topological abelian group, which then acts fibrewise on the
+map θ × Y and hence acts on compatibly Eθ×Y and BDiffθ×Y (Wg). Using this we
+can form the homotopy fibre sequence
+map(Wg, Y )//Y −→ BDiffθ×Y (Wg)//Y −→ BDiffθ(Wg).
+The space map(Wg, Y )//Y is a K(Hn(Wg; Q)⊗W ∨, 1), so there is an identification
+of graded local coefficient systems
+H∗(map(Wg, Y )//Y ; Q) = Λ∗(H ⊗ W[1]).
+This is natural in the vector space W, and scaling by u ∈ Q× acts on Λk(H⊗ W[1])
+by uk. It follows that it acts this way on the kth row of the Serre spectral sequence
+Ep,q
+2
+= Hp(BDiffθ(Wg); Λq(H ⊗ W[1])) ⇒ Hp+q(BDiffθ×Y (Wg)//Y ; Q).
+As the differentials in this spectral sequence must be equivariant for this Q×-action,
+it follows that they must all be trivial.
+Furthermore this action gives a weight
+decomposition of both sides, which identifies
+H∗(BDiffθ(Wg); Λk(H ⊗ W)) ∼= H∗+k(BDiffθ×Y (Wg)//Y ; Q)(k),
+the weight k-subspace.
+To access the latter groups, we use that there is a map
+α : BDiffθ×Y (Wg) −→ Ω∞
+0 (MTθ ∧ Y+)
+
+16
+OSCAR RANDAL-WILLIAMS
+which by the main theorems of [Bol12, RW16, GTMW09] (for 2n = 2) and [GRW18,
+GRW14, GRW17] for (2n ≥ 6) is an isomorphism on cohomology in a stable
+range of degrees. Here MTθ is the Thom spectrum of −θ∗γ2n, so writing u−2n ∈
+H−2n(MTθ; Q) for its Thom class, by the Thom isomorphism we have
+H∗(MTθ; Q) ∼= u−2n · H∗(BSO(2n)⟨n⟩; Q) = u−2n · Q[p⌈n+1
+4
+⌉, . . . , pn−1, e].
+The rational cohomology of Ω∞
+0 (MTθ ∧ Y+) is then given by
+Sym∗([H∗(MTθ; Q) ⊗ Sym∗(W[n + 1])]>0),
+which can be considered as the free (graded-)commutative algebra on the even-
+degree classes κc,w1···wr with c ∈ Q[p⌈ n+1
+4
+⌉, . . . , pn−1, e] and wi ∈ W, modulo lin-
+earity in c and in the wi, and modulo commutativity of the wi. The pullbacks
+of these classes along α we again denote κc,w1···wr, and they may be described
+intrinsically as the fibre integrals πY
+! (c(TπY Eθ×Y ) · ℓ∗(w1 · · · wr)).
+Lemma 3.6. There are unique classes ¯κc,w1···wr ∈ H∗(BDiffθ×Y (Wg)//Y ; Q) which
+pull back to
+�
+I⊔J={1,2,...,r}
+κc,wI ·
+�
+j∈J
+(− 1
+χκe,wj) ∈ H∗(BDiffθ×Y (Wg); Q),
+and in a stable range of degrees H∗(BDiffθ×Y (Wg)//Y ; Q) is the free graded-commutative
+algebra on the classes ¯κc,w1···wr, modulo linearity in c and in the wi, commutativity
+of the wi, and modulo ¯κe,w1 = 0.
+Proof. We wish to apply Lemma 3.5 to the principal Y -bundle
+(3.2)
+BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg)//Y.
+First observe that the fibre inclusion j : Y → BDiffθ×Y (Wg) classifies the Wg-
+bundle pr1 : Y × Wg → Y equipped with the product θ-structure and the map
+ℓ = pr1 : Y × Wg → Y . Thus for any w ∈ W we have
+j∗κe,w = χw ∈ Hn+1(Y ; Q),
+and so (3.2) satisfies the Leray–Hirsch property.
+Lemma 3.5 then describes H∗(BDiffθ×Y (Wg)//Y ; Q) as the equaliser of
+(3.3)
+H∗(BDiffθ×Y (Wg); Q)
+H∗(Y ; Q) ⊗ H∗(BDiffθ×Y (Wg); Q).
+a∗
+1⊗Id
+In a stable range H∗(BDiffθ×Y (Wg); Q) is described in terms of the classes κc,w1···wr,
+so to make use of this equaliser description we must determine how these classes
+pull back along the action map
+a : Y × BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg).
+This map classifies the Wg-bundle Y × πY : Y × Eθ×Y → Y × BDiffθ×Y (Wg)
+equipped with the structure map Y × Eθ×Y
+Y ×ℓ
+→
+Y × Y
+·→ Y .
+As the wi ∈
+W = Hn+1(Y ; Q) are primitive with respect to the coproduct induced by the
+multiplication on Y , we have
+a∗(κc,w1···wr) = (Y × πY )!((1 × c(TπY Eθ×Y )) ·
+r
+�
+i=1
+(wi × 1 + 1 × ℓ∗(wi)))
+=
+�
+I⊔J={1,2,...,r}
+wI × κc,wJ.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+17
+Our goal now is to show that the classes defined by
+¯κc,w1···wr :=
+�
+I⊔J={1,2,...,r}
+κc,wI ·
+�
+j∈J
+(− 1
+χκe,wj) ∈ H∗(BDiffθ×Y (Wg); Q)
+are equalised by the maps (3.3), so by Lemma 3.5Lemma 3.5 descend to unique
+classes of the same name in H∗(BDiffθ×Y (Wg)//Y ; Q). To see this, we calculate
+using the formula above that
+a∗(¯κc,w1···wr) =
+�
+I⊔J={1,2,...,r}
+� �
+S⊔T =I
+wS × κc,wT
+�
+· (−1)|J| �
+j∈J
+(wj × 1 + 1
+χ1 × κe,wj)
+=
+�
+S⊔T ⊔U⊔V ={1,2,...,r}
+(−1)|U|wS⊔U ×
+�
+κc,wT ·
+�
+v∈V
+(− 1
+χκe,wv)
+�
+.
+For each A ⊆ {1, 2, . . ., r} the coefficient of wA is
+
+ �
+U⊆A
+(−1)|U|
+
+
+
+
+�
+T ⊔V ={1,...,r}\A
+κc,wT ·
+�
+v∈V
+(− 1
+χκe,wv)
+
+
+and �
+U⊆A(−1)|U| vanishes if A ̸= ∅, and is 1 if A = ∅ (it is the binomial expansion
+of (1 − 1)|A|), which shows that a∗(¯κc,w1···wr) = 1 × ¯κc,w1···wr as required.
+Finally, that these classes (except ¯κe,w1 = 0) freely generate the Q-algebra
+H∗(BDiffθ×Y (Wg)//Y ; Q) in a stable range follows from the fact that the κc,w1···wr
+freely generate H∗(BDiffθ×Y (Wg); Q) in a stable range, together with the observa-
+tion that ¯κc,w1···wr ≡ κc,w1···wr modulo the ideal generated by classes κe,w and the
+Leray–Hirsch property again.
+□
+Let us provide a “fibre-integral” interpretation of the classes we have just con-
+structed. Consider the map of principal Y -bundles
+Y
+Eθ×Y
+Eθ×Y //Y
+Y
+BDiffθ×Y (Wg)
+BDiffθ×Y (Wg)//Y.
+i
+πY
+πY //Y
+j
+The composition ℓ ◦ i : Y → Y is the identity, so i∗ℓ∗(w) = w ∈ Hn+1(Y ; Q). We
+showed in the proof above that j∗κe,w = χw ∈ Hn+1(Y ; Q), so in particular both
+these principal Y -bundles satisfy the Leray–Hirsch property. Together these give
+that
+i∗(ℓ∗(w) − 1
+χ(πY )∗κe,w) = 0.
+As Y is n-connected it follows from the Serre spectral sequence that there exists a
+unique class ¯ℓ∗(w) ∈ Hn+1(Eθ×Y //Y ; Q) which pulls back to ℓ∗(w) − 1
+χ(πY )∗κe,w.
+Lemma 3.7. We have
+¯κc,w1···wr = (πY //Y )!(c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) ∈ H∗(BDiffθ×Y (Wg)//Y ; Q).
+Proof. As the lower of the above principal Y -bundles satisfies the Leray–Hirsch
+property, this identity may be verified after pulling back to BDiffθ×Y (Wg).
+In
+H∗(Eθ×Y ; Q) we have ¯ℓ∗(w) = ℓ∗(w) − 1
+χ(πY )∗κe,w, so expanding out gives
+(πY )!(c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) = (πY )!(c ·
+r
+�
+i=1
+(ℓ∗(wi) − 1
+χ(πY )∗κe,wi))
+=
+�
+I⊔J={1,2,...,r}
+κc,wI ·
+�
+j∈J
+(− 1
+χκe,wj)
+
+18
+OSCAR RANDAL-WILLIAMS
+as required.
+□
+The classes ¯κc,w1···wr provide an isomorphism
+Sym∗
+�[H∗(MTθ; Q) ⊗ Sym∗(W[n + 1])]>0
+u−2n · e ⊗ W[n + 1]
+�
+−→ H∗(BDiffθ×Y (Wg)//Y ; Q)
+in a stable range, natural in W, which with the discussion above gives an identifi-
+cation of graded vector spaces
+H∗(BDiffθ(Wg); Λ∗(H ⊗ W[1])) ∼= Sym∗
+�[H∗(MTθ; Q) ⊗ Sym∗(W[n + 1])]>0
+u−2n · e ⊗ W[n + 1]
+�
+natural in W.
+Just as in the proof of [KRW20b, Theorem 3.15], and using its notation, this
+implies that there is a natural transformation
+(3.4)
+Pbis(−, V)≥0 ⊗ det⊗n −→ H∗(BDiffθ(Wg); H⊗−)
+of lax symmetric monoidal functors FB → Gr(Q-mod) which is an isomorphism in a
+stable range, where P(−, V)≥0 → Pbis(−, V)≥0 is the quotient by those partitions
+containing a part of size 1 labelled by e ∈ V2n. Assigning to a labelled part the
+corolla with that label gives a natural transformation
+(3.5)
+Pbis(−, V)≥0 ⊗ det⊗n −→ Graphθ(−)g,
+of lax symmetric monoidal functors FB → Gr(Q-mod), and we claim that using this
+(3.4) factors through the map ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg); H⊗−). Assuming
+this claim for now, observe that using the contraction relations in Definition 3.1 (iv)
+(d′′′) to contract all edges shows that (3.5) is surjective, which with the fact that
+(3.4) is an isomorphism in a stable range will show that the map ¯κ is an isomorphism
+in a stable range too (as well as the map (3.5)).
+It remains to show the factorisation, i.e. that the map (3.4) sends a part of size
+a labelled by c ∈ V to the class κ¯εac. We again proceed as in the relevant step of
+the proof of [KRW20b, Theorem 3.15]. There is a fibration sequence
+map(Wg, Y ) −→ Eθ×Y −→ Eθ
+and so, taking homotopy orbits for the fibrewise Y -action, a fibration sequence
+map(Wg, Y )//Y −→ Eθ×Y //Y −→ Eθ.
+Again by functoriality in W the associated Serre spectral sequence collapses to
+identify the weight decomposition as
+H∗(Eθ; Λk(H ⊗ W)) ∼= H∗+k(Eθ×Y //Y ; Q)(k).
+Given the description in Lemma 3.7 we must show that the map
+¯ℓ(−) : W −→ Hn+1(Eθ×Y //Y ; Q)(1) ∼= Hn(Eθ; H) ⊗ W
+is given by w �→ ¯ε ⊗ w, which is the analogue of [KRW20b, Claim 3.16]. As it is
+natural in the vector space W it must certainly be given by ¯ℓ(w) = x ⊗ w for some
+x ∈ Hn(Eθ; H), and we must show that x = ¯ε. That the restriction of x to the
+fibre Wg of π : Eθ → BDiffθ(Wg) is given by coevaluation may be done precisely
+as in [KRW20b, Claim 3.16]. By the characterisation of ¯ε it remains to check that
+1
+χ(πY )!(e · ¯ℓ∗(w)) = 0 ∈ Hn+1(BDiffθ×Y (Wg)//Y ; Q).
+By the Leray–Hirsch property this may be checked after pulling back to BDiffθ×Y (Wg),
+but as ¯ℓ∗(w) = ℓ∗(w) − 1
+χ(πY )∗κe,w ∈ Hn+1(Eθ×Y ; Q) by definition, the vanishing
+is immediate.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+19
+3.4. Comparisons. There are natural maps
+BDiff(Wg, D2n)
+BDiffθ(Wg, ∗)
+BDiffθ(Wg)
+BDiff(Wg, D2n)
+BDiff+(Wg, ∗)
+BDiff+(Wg)
+a
+b
+c
+d
+e
+f
+which each induce maps on H∗(−; H⊗S). There are corresponding maps of spaces
+of graphs
+Graph1(−)g
+Graphθ
+∗(−)g
+Graphθ(−)g
+Graph1(−)g
+Graph∗(−)g
+Graph(−)g
+a∗
+b∗
+e∗
+c∗
+d∗
+f ∗
+given as follows. The maps c∗ and d∗ are induced by the projections W → V. The
+maps a∗ and e∗ are induced by applying the augmentations V → Q and W → Q
+to the second tensor factor. The maps b∗ and f ∗ are more subtle, as they involve
+converting between blue graphs and red graphs, via the formula of Proposition 2.7.
+Graphically it is given by
+•
+�→
+•
+− 1
+χ(
+•
+•
+e
+•
+•
+e
+•
+•
+e
++
++
+)
+with certain orderings.
+The maps b and f are also oriented Wg-bundles, so they also induce fibre-
+integration maps b! and f! on cohomology. These are b∗- and f ∗-linear respectively,
+so are determined by the maps (of degree −2n)
+b! : V −→ Graphθ(−)g
+f! : W −→ Graph(−)g
+which each send a monomial c in pi’s and e to the graph given by a single vertex
+labelled by c.
+4. Cohomology of Torelli groups
+The isomorphisms provided by Theorem 3.4 can be converted into information
+about the spaces
+BTor(Wg, D2n), BTorθ(Wg, ∗), BTor+(Wg, ∗), BTorθ(Wg), BTor+(Wg)
+just as [KRW20a, Theorem 4.1] is deduced from [KRW20a, Theorem 3.15]. Let us
+give the definition of these spaces and formulate the result: the following is largely
+a reminder of some points from [KRW20a], and we do not spell out all details again.
+The group Diff+(Wg) acts on Hn(Wg; Z) preserving the nondegenerate (−1)n-
+symmetric intersection form λ : Hn(Wg; Z) ⊗ Hn(Wg; Z) → Z. This provides a
+homomorphism
+αg : Diff+(Wg) −→ Gg :=
+�
+Sp2g(Z)
+if n is odd,
+Og,g(Z)
+if n is even.
+This map is not always surjective, but its image is a certain finite-index subgroup
+G′
+g ≤ Gg, an arithmetic group associated to the algebraic group Sp2g or Og,g. This
+subgroup has been determined by Kreck [Kre79]: it is the whole of Gg if n is even
+or n = 1, 3, 7, and otherwise is the subgroup Spq
+2g(Z) ≤ Sp2g(Z) of those matrices
+which preserve the standard quadratic refinement (of Arf invariant 0).
+
+20
+OSCAR RANDAL-WILLIAMS
+We define Tor+(Wg) to be the kernel of this homomorphism, and Tor+(Wg, ∗)
+and Tor(Wg, D2n) to be the kernel of its restriction to the subgroups Diff+(Wg, ∗)
+and Diff(Wg, D2n) respectively (these restrictions still have image G′
+g). Further-
+more, we define
+BTorθ(Wg) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg)
+BTorθ(Wg, ∗) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg, ∗),
+where Bun+(T Wg, θ∗γ2n) ⊂ Bun(T Wg, θ∗γ2n) consists of the orientation-preserving
+bundle maps (for some choice of orientation of θ∗γ2n that we make once and for
+all). By the discussion at the beginning of Section 3 the spaces Bun+(T Wg, θ∗γ2n)
+are path-connected, so each of the BTor’s we have defined are principal G′
+g-bundles
+over the corresponding BDiff’s. In particular, their rational cohomologies are both
+Q-algebras and G′
+g-representations, and we will describe them as such in a stable
+range. Before doing so, we recall that the work of Borel identifies
+H∗(G′
+g; Q) =
+�
+Q[σ2, σ6, σ10, . . .]
+if n is odd,
+Q[σ4, σ8, σ12, . . .]
+if n is even.
+in a stable range of degrees, where σ4i−2n may be chosen so that it pulls back to the
+Miller–Morita–Mumford class κLi ∈ H4i−2n(BDiff+(Wg; Q) associated to the ith
+Hirzebruch L-class. In particular the κLi vanish in the cohomology of BTor+(Wg).
+Let us write H(g) := Hn(Wg; Q), which is the standard representation of G′
+g.
+Pulled back from BDiff+(Wg) to BTor+(Wg) the coefficient system H is canonically
+trivialised, but has an action of G′
+g: it can be identified with the dual H(g)∨. The
+edge homomorphism of the Serre spectral sequence
+(4.1)
+H∗(BDiff+(Wg); H⊗S) −→
+�
+H∗(BTor+(Wg); Q) ⊗ (H(g)∨)⊗S�G′
+g
+allows us to consider the modified twisted Miller–Morita–Mumford classes ¯κεSc as
+providing G′
+g-equivariant homomorphisms
+¯κc : H(g)⊗S −→ Hn(|S|−2)+|c|(BTor+(Wg); Q).
+The identities from the modified contraction formula correspond to identities
+among these maps: this will give relations analogous to [KRW20b, Section 5.2],
+which we will spell out after the proof of Theorem 4.1 below. First we explain how
+these relations can be organised in a categorical way, as follows.
+Considering (4.1) as a natural transformation of functors on (s)Br2g, we may
+precompose it with the map
+¯κ : Graphg(−) −→ H∗(BDiff+(Wg); H⊗−)
+(which is an isomorphism in a stable range for n ̸= 2 by Theorem 3.4). This gives
+G′
+g-equivariant maps H(g)⊗S ⊗ Graphg(S) → H∗(BTor+(Wg); Q) which assemble
+to a map
+K∨ ⊗(s)Br2g Graphg(−) −→ H∗(BTor+(Wg); Q)
+out of the coend, where K : (s)Br2g → Rep(G′
+g) sends S to H(g)⊗S. The domain
+obtains a graded-commutative Q-algebra structure coming from the lax symmetric
+monoidality of Graphg(−) and strong symmetric monoidality of K(−). Theorem
+4.1 below will say that this is surjective in a stable range, with kernel the ideal
+generated by the κLi, but before stating it we explain a simplification.
+Let us write i : d(s)Br → (s)Br2g for the inclusion of the downward (signed)
+Brauer category. Thus subcategory is independent if g, as no circles can be created
+by composing morphisms in the downward Brauer category. Write Graph1(−)′ ⊂
+i∗Graph1(−)g for the subfunctor where we forbid bivalent vertices labelled by 1 ∈ V
+both of whose half-edges are legs; similarly, this functor is independent of g. Like
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+21
+just after [KRW20b, Proposition 3.11], Graph1(−)g is then the left Kan extension
+i∗Graph1(−)′ of Graph1(−)′ along i. We similarly define Graphθ
+∗(−)′, Graph∗(−)′,
+Graphθ(−)′, and Graph(−)′, whose left Kan extensions again recover the original
+functors. The following is the analogue of [KRW20b, Theorem 4.1].
+Theorem 4.1. There are G′
+g-equivariant ring homomorphisms
+i∗(K∨) ⊗d(s)Br Graph1(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTor(Wg, D2n); Q)
+(i)
+i∗(K∨) ⊗d(s)Br Graphθ
+∗(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTorθ(Wg, ∗); Q)
+(ii)
+i∗(K∨) ⊗d(s)Br Graph∗(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTor+(Wg, ∗); Q)
+(iii)
+i∗(K∨) ⊗d(s)Br Graphθ(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTorθ(Wg); Q)
+(iv)
+i∗(K∨) ⊗d(s)Br Graph(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTor+(Wg); Q)
+(v)
+which for 2n ≥ 6 are isomorphisms in a stable range of degrees.
+If 2n = 2 then, in a stable range of degrees and assuming that the target is
+finite-dimensional in degrees ∗ < N for all large enough g, these maps are iso-
+morphisms onto the maximal algebraic subrepresentations in degrees ∗ ≤ N, and
+monomorphisms in degrees ∗ ≤ N + 1.
+Proof. By the main theorem of [KRW20a], as long as 2n ≥ 6 the G′
+g-representations
+Hi(BTor(Wg, D2n); Q) are algebraic. Using the inheritance properties for algebraic
+representations from [KRW20a, Theorem 2.2], the Serre spectral sequences for the
+homotopy fibre sequences
+BTor(Wg, D2n) −→BTor+(Wg, ∗) −→ BSO(2n)
+BTor(Wg, D2n) −→BTorθ(Wg, ∗) −→ BSO(2n)⟨n⟩
+show that the cohomology groups of BTor+(Wg, ∗) and BTorθ(Wg, ∗) are also al-
+gebraic G′
+g-representations, and the same for the homotopy fibre sequences
+Wg −→BTor+(Wg, ∗) −→ BTor+(Wg)
+Wg −→BTorθ(Wg, ∗) −→ BTorθ(Wg)
+show that the cohomology groups of BTor+(Wg) and BTorθ(Wg) are algebraic
+G′
+g-representations too.
+Using this algebraicity property, case (i) is precisely [KRW20b, Theorem 4.1],
+using that by [KRW20b, Proof of Theorem 5.1] Graph1(−)g is isomorphic to the
+functor P(−, V)≥0⊗det⊗n. The other cases follow in the same way, using [KRW20b,
+Proposition 2.16], from Theorem 3.4, with one elaboration which we describe below.
+The addendum in the case 2n = 2 is precisely as in [KRW20b, Theorem 4.1].
+The elaboration comes when verifying the first hypothesis of [KRW20b, Lemma
+4.3], which in case (v) for example requires us to know that H∗(BDiff+(Wg); H⊗S)
+is a free H∗(G′
+g; Q)-module in a stable range. But by transfer H∗(BDiff+(Wg); H⊗S)
+is a summand of H∗(BDiff+(Wg, ∗); H⊗S) (as H∗(G′
+g; Q)-modules), and similarly
+with θ-structures, so cases (ii) and (iii) imply cases (iv) and (v).
+In the other
+hand in case (iii) for example we have discussed in the proof of Theorem 3.4 the
+degeneration of the Serre spectral sequence in a stable range, giving
+gr(H∗(BDiff+(Wg, ∗); H⊗S)) ∼= H∗(BSO(2n); Q) ⊗ H∗(BDiff(Wg, D2n); H⊗S).
+
+22
+OSCAR RANDAL-WILLIAMS
+The Serre filtration is one of H∗(G′
+g; Q)-modules, so as the associated graded is a
+free H∗(G′
+g; Q)-module in a stable range (because H∗(BDiff(Wg, D2n); H⊗S) is the
+case treated in [KRW20b, Theorem 4.1]), it follows that H∗(BDiff+(Wg, ∗); H⊗S)
+is too. The same argument applies in case (ii).
+□
+This quite categorical description can be used to get a more down-to-earth pre-
+sentation for these cohomology rings: in case (v) this is the presentation we have
+recorded in Theorem A. This is deduced just as in [KRW20b, Section 5], though
+most of the work has been done as we have already expressed things in terms of
+graphs. As in [KRW20b, Section 5.4] this is not the smallest possible presentation:
+it can be simplified by manipulating graphs; we leave the details to the interested
+reader.
+5. The case 2n = 2
+Although Theorem 4.1 is only known to hold in a limited range of degrees in the
+case 2n = 2 (N = 2 is currently the best known constant for g ≥ 3, using the work
+of Johnson [Joh85]), Theorem 3.4 does hold in a range of cohomological degrees
+tending to infinity with g. In this case our discussion is closely related to the work
+of Kawazumi and Morita [Mor96, KM96, KM01], and in this section we we take
+the opportunity to revisit that work from our perspective. Throughout this section
+we assume that g ≥ 2, so that χ(Wg) = 2 − 2g ̸= 0.
+In terms of Kawazumi and Morita’s notation we have
+Mg := π0(Diff+(Wg))
+Mg,∗ := π0(Diff+(Wg, ∗))
+Mg,1 := π0(Diff+(Wg, D2)).
+Under our assumption g ≥ 2 the groups Diff+(Wg), Diff+(Wg, ∗), and Diff+(Wg, D2)
+all have contractible path-components, so the group cohomology of Mg is the co-
+homology of BDiff+(Wg), and so on. Theorem 3.4 gives a natural transformation
+¯κ : Graph(−)g −→ H∗(Mg; H⊗−)
+of functors sBr2g → Gr(Q-mod), which is an isomorphism in a stable range of
+degrees. Note that in this case H∗(BSO(2); Q) = Q[e] so V = W = Q[e] and there
+is no difference between the tangential structure θ and an orientation. In particular
+if we denote by Γi ∈ Graph(∅) the graph with a single vertex, no edges, and labelled
+by ei+1, then ¯κ(Γi) = κi ∈ H2i(Mg; Q) is the usual Miller–Morita–Mumford class4.
+Our goal in Sections 5.1–5.4 is to analyse Graph(−) in several ways, making
+contact with the work of Kawazumi and Morita mentioned above as well as work
+of Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].
+5.1. Reduction to corollas. The possible labels for the vertices of graphs in
+Graph(S) are powers of the Euler class e. Given any graph we may iteratedly apply
+the modified contraction formula to write it as a linear combination of graphs with
+fewer edges, and hence any graph is equivalent to a linear combination of graphs
+with no edges: these are disjoint unions of corollas. Of these, by definition of Graph:
+the 0-valent corolla labelled by e is equal to the scalar χ, the 1-valent corolla labelled
+by 1 ∈ V is trivial, and the 1-valent corolla labelled by e ∈ V is trivial. Define a
+labelled partition of a finite set S to be a partition {Sα}α∈I of S into (possibly
+empty) subsets and a label enα for each part, such that
+(i) If |Sα| = 0 then nα ≥ 2,
+(ii) If |Sα| = 1 then nα ≥ 1.
+4Our κi is denoted ei in the work of Kawazumi and Morita.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+23
+We give a part (Sα, nα) degree 2nα + |Sα| − 2, and a labelled partition the degree
+given by the sums of the degrees of its parts. Similarly to the proof of Theorem 3.4
+(iv) (particularly around equation (3.5)), let Pbis(S, V)≥0 denote the free Q[χ±1]-
+module with basis the set of labelled partitions of S. Assigning to a labelled part
+(Sα, enα) the corolla with legs Sα and label enα defines a map
+(5.1)
+Pbis(S, V)≥0 ⊗ det QS −→ Graph(S),
+natural in S with respect to bijections.
+Lemma 5.1. The map (5.1) is an isomorphism.
+Proof. It is surjective, as explained above, by repeatedly applying the modified
+contraction formula to express a graph in terms of graphs without edges.
+If it were not injective then it would have some nontrivial Q[χ±1]-linear com-
+bination of labelled partitions in its kernel, of a given degree d, and this would
+remain a nontrivial Q-linear combination of labelled partitions when specialised to
+χ = 2 − 2g for all g ≫ 0 (as a Laurent polynomial in χ has finitely-many roots).
+But in the proof of Theorem 3.4 (iv), in the discussion after equation (3.5), it is
+explained that when specialised to χ = 2 − 2g this map is an isomorphism in a
+range of degrees tending to infinity with g; for large enough g the degree d will be
+in this stable range, a contradiction.
+□
+In particular, for the graphs Γi described above there is an isomorphism
+(5.2)
+Q[χ±1][Γ1, Γ2, . . .] ∼= Graph(∅).
+5.2. Reduction to trivalent graphs without labels. In this section we will
+prove the following.
+Theorem 5.2. Using the modified contraction formula any marked oriented graph
+is equivalent to a Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1]-linear combination of
+trivalent graphs with all vertices labelled by 1 ∈ V0.
+Let Graphtri(S) ≤ Graph(S) denote the sub-Q[χ±1]-module spanned by those
+marked oriented graphs which are trivalent and all of whose labels are 1 ∈ V.
+Corollary 5.3. The monomorphism i : Graphtri(−) → Graph(−) becomes an iso-
+morphism upon inverting χ − 2, χ − 3, and χ − 4. In particular Graphtri(−)g =
+Graph(−)g.
+Remark 5.4 (2-valent vertices labelled by 1). Using the relation
+λ2,3(κ¯ε1,2κ¯ε3,...,nc) = κ¯ε1,3,...,nc
+we can always remove 2-valent vertices labelled by 1. It is sometimes convenient
+when writing formulas for 3-valent graphs to also allow 2-valent vertices labelled
+by 1: we allow ourselves to do so, noting that the above can always be used to
+eliminate the 2-valent vertices.
+Proof of Theorem 5.2. As a matter of notation we will formally manipulate modi-
+fied twisted Miller–Morita–Mumford classes, but this is equivalent to manipulating
+marked oriented graphs. Rearranging the first contraction formula gives
+(5.3)
+κ¯εaeb =
+χ
+χ−2
+�
+λ1,2κ¯ε2+aeb−1 −
+1
+χ2 κe2κ¯εaeb−1
+�
+.
+Rearranging the second contraction formula gives
+κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) −
+1
+χ2 (κe2 · κ¯εa · κ¯εb) + 1
+χ(κ¯εae · κ¯εb + κ¯εa · κ¯εbe)
+
+24
+OSCAR RANDAL-WILLIAMS
+and using (5.3) to eliminate the Euler classes from the last two terms gives
+κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) −
+1
+χ2 (κe2 · κ¯εa · κ¯εb)
++
+1
+χ−2((λ1,2(κ¯ε2+a) −
+1
+χ2 κe2κ¯εa) · κ¯εb + κ¯εa · (λ1,2(κ¯ε2+b) −
+1
+χ2 κe2κ¯εb))
+= λa+1,a+2(κ¯εa+1 · κ¯ε1+b) +
+1
+χ−2 ((λ1,2(κ¯ε2+a) · κ¯εb + κ¯εa · λ1,2(κ¯ε2+b))
+−
+1
+χ(χ−2)κe2 · κ¯εa · κ¯εb.
+It suffices to show that each corolla κ¯εaeb may be represented by a linear combi-
+nation of trivalent graphs. By Example 2.8 the class κe2 may be represented by a
+trivalent graph (after inverting χ − 3) so by iteratedly applying (5.3) it suffices to
+show that each κ¯εn can too. By Remark 5.4 we may as well show that classes can
+be represented by 2- and 3-valent graphs. To get started we have κ¯ε = 0 as it has
+negative degree.
+Consider the class λ2,5λ3,4(κ¯ε1,2,3 ·κ¯ε4,5,6). Using the form of the relations above,
+which avoid creating Euler classes, this is
+λ2,5(κ¯ε1,2,5,6 −
+1
+χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6 + κ¯ε1,2λu,v(κ¯εu,v,5,6)) +
+1
+χ(χ−2)(κe2κ¯ε1,2κ¯ε5,6))
+= λ2,5(κ¯ε1,2,5,6) −
+2
+χ−2λu,v(κ¯εu,v,1,6) +
+1
+χ(χ−2)κe2κ¯ε1,6
+= χ−4
+χ−2λ2,5(κ¯ε1,2,5,6) +
+1
+χ(χ−2)κe2κ¯ε1,6
+Renumbering legs and rearranging, this shows that λ1,2(κ¯ε4) may be represented
+by 2- and 3-valent graphs.
+Applied with (a, b) = (2, 2) the second relation gives
+κ¯ε4 = λ3,4(κ¯ε3 · κ¯ε3) +
+1
+χ−2 ((λ1,2(κ¯ε4) · κ¯ε2 + κ¯ε2 · λ1,2(κ¯ε4)) −
+1
+χ(χ−2)κe2 · κ¯ε2 · κ¯ε2,
+which with the above shows that κ¯ε4 may be represented by 2- and 3-valent graphs.
+Similarly to the above, consider λ2,5λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6,7), which is
+λ2,5(κ¯ε1,2,5,6,7 +
+1
+χ(χ−2)κe2κ¯ε1,2κ¯ε5,6,7 −
+1
+χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6,7 + κ¯ε1,2λu,v(κ¯εu,v,5,6,7)))
+= λ2,5(κ¯ε1,2,5,6,7) +
+1
+χ(χ−2)κe2κ¯ε1,6,7 −
+1
+χ−2(λ2,5λu,v(κ¯εu,v,1,2κ¯ε5,6,7) + λu,v(κ¯εu,v,1,6,7))
+= χ−3
+χ−2λ2,5(κ¯ε1,2,5,6,7) +
+1
+χ(χ−2)κe2κ¯ε1,6,7 −
+1
+χ−2λ2,5λu,v(κ¯εu,v,1,2)κ¯ε5,6,7.
+Renumbering legs and rearranging, this shows that λ1,2(κ¯ε5) may be represented by
+2-, 3-, and 4-valent graphs; with the above it follows that it can also be represented
+by 2- and 3-valent graphs.
+Applied with (a, b) = (2, 3) the second relation gives
+κ¯ε5 = λ3,4(κ¯ε3 · κ¯ε4) +
+1
+χ−2 ((λ1,2(κ¯ε4) · κ¯ε3 + κ¯ε2 · λ1,2(κ¯ε5)) −
+1
+χ(χ−2)κe2 · κ¯ε2 · κ¯ε3,
+so it follows that κ¯ε5 may be represented by 2- and 3-valent graphs.
+If n ≥ 6 then we can write n = a + b with a, b ≥ 3, so a + 2, b + 2 < n and so the
+second relation expresses κ¯εn in terms of κ¯εm’s with m < n. Thus all κ¯εn’s may be
+represented by 2- and 3-valent graphs as required.
+□
+It is worth observing that we have the relation
+(5.4)
+λ1,2(κ¯ε3) = χ−2
+χ κ¯εe +
+1
+χ2 κe2κ¯ε = 0,
+using that κ¯εe = 0 (by definition) and that κ¯ε = 0 (as it has negative degree). This
+means that any graph having a trivalent vertex with a loop is trivial in Graph(−).
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+25
+5.3. A remark on orderings. A curious normalisation is possible when consider-
+ing trivalent graphs, allowing one to neglect the orderings of vertices, of half-edges,
+and the orientations of edges. In [Mor96, KM96, KM01] this is implemented ab
+initio and (marked) oriented graphs play no role. Let us explain this normalisation,
+extended to trivalent graphs with legs.
+A trivalent graph ˜Γ with legs S consists of a set V of vertices, a set H of half-
+edges, a 3-to-1 map a : H → V recording to which vertex each half-edge is incident,
+and an unordered matching µ on H ⊔ S recording which half-edges span an edge,
+and which half-edges are connected to which legs.
+Given a trivalent graph ˜Γ = (V, H, a : H → V, µ) with legs S, we may choose an
+ordering of V and choose an ordering of H such that a : H → V is weakly monotone
+(equivalently, choose an ordering of the half-edges incident at each vertex). We also
+choose an ordering of S.
+There is an induced ordering of H ⊔ S by putting ⃗S
+after ⃗H, and we form an ordered matching m of H ⊔ S by taking those pairs
+(a, b) with a < b and {a, b} ∈ µ. Using this we form an oriented trivalent graph
+Γchoice = (⃗V , ⃗H, a : H → V, m), depending on these choices. The normalisation is
+as follows. Let x1 < x2 < x3 < x4 < . . . < x2k ∈ H ⊔ S be the total order on
+H ⊔ S, and let a1 < b1, . . . , ak < bk be the ordered pairs which span an edge, with
+a1 < a2 < . . . < ak ∈ H ⊔ S. Then there is a bijection given by
+ρ :=
+� a1 b1 a2 b2 a3 b3 ···
+ak
+bk
+x1 x2 x3 x4 x5 x6 ··· x2k−1 x2k
+�
+and we define Γ := sign(ρ) · Γchoice.
+Claim. As long as ˜Γ has no vertices with loops, the element Γ does not depend
+on the choice of ordering of V or H, and depends on the ordering of S precisely as
+the sign representation.
+In particular if we set5
+Graphundec(S) := Q[χ±1][˜Γ trivalent graph with legs S]/(graphs with loops)
+then the Claim together with the relation (5.4) provides an epimorphism
+Φ : Graphundec(S) ⊗ det QS −→ Graphtri(S)
+of Q[χ±1]-modules, natural with respect to bijections in S. This can be extended to
+a natural transformation of functors on sBrχ by letting an ordered matching (a, b)
+of elements of S act by adding an edge to the trivalent graph connecting a and b,
+and contracting the determinant by a ∧ b. Doing so might create a circle with no
+vertices, which should be replaced by the scalar χ − 2.
+Proof of Claim. If (h1, h2, h3) are the half-edges incident at a vertex v and we
+change their ordering to (hσ(1), hσ(2), hσ(3)) giving Γ′
+choice, then (under the assump-
+tion that Γ does not have loops) the relative ordering of half-edges forming an
+edge has not changed, so m′ = m. Thus Γ′
+choice = sign(σ) · Γchoice. On the other
+hand ρ′ is obtained from ρ by postcomposing with σ, and precomposing with a
+permutation which permutes some (ai < bi)’s, which is an even permutation. Thus
+sign(ρ′) = sign(σ) · sign(ρ), so Γ′ = Γ.
+Suppose a vertex v1 has half edges (h1
+1, h1
+2, h1
+3) and v2 has half edges (h2
+1, h2
+2, h2
+3),
+and v1 < v2 ∈ ⃗V are adjacent in the ordering on V , and consider transposing the
+ordering of these vertices. For edges between a u < v1 and a vi or between a vi and
+a u > v2 the relative ordering of their half-edges does not change. Edges between
+v1 and v2 have the relative ordering of their half-edges reversed. Thus if there are
+N such edges we have Γ′
+choice = (−1)1+N ·Γchoice. But the permutation ρ is changed
+5In [Mor96, KM96, KM01] they restrict to “trivalent graphs without loops”, however we find
+it more natural to allow loops but set graphs with a loop to zero.
+
+26
+OSCAR RANDAL-WILLIAMS
+by permuting (h2
+1, h2
+2, h2
+3) past (h1
+1, h1
+2, h1
+3), which has sign −1, and N transpositions
+(aibi), which has sign (−1)N. Thus again Γ′ = Γ.
+Finally, changing the order of S by a permutation τ changes ρ by postcomposition
+with τ, so acts as sign(τ).
+□
+Example 5.5. For the ordering of vertices and half-edges corresponding to the
+theta-graph in Example 2.8 the associated permutation is ρ = (1)(235)(46) which
+is odd, so the undecorated theta-graph yields χ−3
+χ κe2. This is precisely minus the
+evaluation of βΓ2 on [KM01, p. 39] (unfortunately the theta-graph is denoted Γ2 in
+that paper). This minus comes from the use of a different sign convention, see the
+discussion at [KRW20b, top of p. 33].
+5.4. Relations among trivalent graphs. The modified contraction formula de-
+scribes relations among graphs involving contracting an edge, but this necessarily
+involves graphs with vertices of different valencies. In Theorem 5.2 we have ex-
+plained that, in the case of surfaces, all graphs may be expressed purely in terms of
+trivalent graphs: one may ask what relations among trivalent graphs Γ are imposed
+by the contraction formula.
+For the unmodified contraction formula discussed in [KRW20b], the answer is
+that it imposes the “I = H” relation among trivalent graphs: this is because
+both the I- and H-graphs admit contractions to the X-graph. Furthermore, as all
+connected trivalent graphs with the same number of legs and of the same genus are
+equivalent under the “I = H” relation, and the contraction formula never changes
+the genus or number of legs, there are no further relations.
+In the setting of the modified contraction formula discussed here it is more
+complicated. It is best given in the setting of undecorated trivalent graphs.
+Theorem 5.6. After inverting χ−2, χ−3, and χ−4, undecorated trivalent graphs
+which differ locally by
+(IHmod)
+=
++
+1
+(χ−4)(3−χ)(
+−
+)
++
+1
+χ−4(
++
+−
+−
+)
+give the same elements in Graphtri[(χ − 2)−1, (χ − 3)−1, (χ − 4)−1].
+Proof. We establish this relation in Graphtri({a, b, c, d})⊗detQ{a,b,c,d}, and it then
+follows in general using functoriality on the signed Brauer category. We order the
+legs as a < b < c < d.
+(i)
+a
+b
+c
+d
+1
+3
+5
+6 2
+4
+(ii)
+a
+b
+c
+d
+1 2
+6
+3 4
+5
+Figure 3. Some marked graphs.
+Consider first the H-shaped graph shown in Figure 3 (i), with the depicted names
+of half edges, ordered as 3 < 1 < 5 < 6 < 2 < 4. Its corresponding permutation is
+� 3 c 1 a 5 6 2 b 4 d
+3 1 5 6 2 4 a b c d
+�
+which is even. Thus this ordering data represents the underlying
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+27
+undecorated H-shaped trivalent graph. Ignoring for now the matchings to the legs
+(which are given by matching 1 with a, 2 with b, and so on), it corresponds to
+λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4). Using the form of the relations which avoid creating Euler
+classes from the proof of Theorem 5.2 we have
+λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4) = κ¯ε3,1,2,4 +
+1
+χ(χ−2)κe2κ¯ε3,1κ¯ε2,4
+−
+1
+χ−2(λu,v(κ¯εu,v,3,1)κ¯ε2,4 + κ¯ε3,1λu,v(κ¯εu,v,2,4)).
+Consider now the I-shaped graph shown in Figure 3 (ii), with the depicted names
+of the half-edges, ordered as 4 < 3 < 5 < 6 < 1 < 2. Its corresponding permutation
+is
+� 4 d 3 c 5 6 1 a 2 b
+4 3 5 6 1 2 a b c d
+�
+which is odd. Thus this ordering data represents minus the
+underlying undecorated I-shaped trivalent graph. Ignoring again the matchings to
+the legs, it corresponds to
+λ5,6(κ¯ε4,3,5 · κ¯ε6,1,2) = κ¯ε4,3,1,2 +
+1
+χ(χ−2)κe2κ¯ε4,3κ¯ε1,2
+−
+1
+χ−2(λu,v(κ¯εu,v,4,3)κ¯ε1,2 + κ¯ε4,3λu,v(κ¯εu,v,1,2)).
+The sum of these two expressions therefore represents the image under Φ of
+the difference H − I of the underlying undecorated trivalent graphs. Furthermore,
+κ¯ε4,3,1,2 = −κ¯ε3,1,2,4 so these terms cancel.
+From the proof of Theorem 5.2 we have the identity
+λu,v(κ¯εu,v,s,t) = χ−2
+χ−4λi,jλk,l(κ¯εs,i,k · κ¯εl,j,t) −
+1
+χ(χ−4)κe2κ¯εs,t,
+expressing terms of the form λu,v(κ¯εu,v,s,t) in terms of (2- and) 3-valent vertices.
+Applying it to the sum of the two expressions above, and collecting terms, therefore
+gives
+Φ(H − I) =
+1
+χ(χ−4)κe2�
+κ¯ε3,1κ¯ε2,4 + κ¯ε4,3κ¯ε2,1�
+−
+1
+χ−4
+�
+λi,jλk,l(κ¯ε3,i,k · κ¯εl,j,1)κ¯ε2,4 + κ¯ε3,1λi,jλk,l(κ¯ε2,i,k · κ¯εl,j,4)
+λi,jλk,l(κ¯ε4,i,k · κ¯εl,j,3)κ¯ε1,2 + κ¯ε4,3λi,jλk,l(κ¯ε1,i,k · κ¯εl,j,2)
+�
+.
+Using that κe2 = Φ(
+χ
+χ−3Θ) and carefully putting the graphs corresponding to the
+other terms into the normal form of Section 5.3 gives the identity in the statement
+of the theorem.
+□
+Our relation IHmod is graphically identical to the relation called IHbis
+0
+by
+Akazawa [Aka05, p. 100] and in the corrigendum [GN07] to the paper of Garo-
+ufalidis and Nakamura [GN98]. In those papers it is emphasised that IHbis
+0
+means
+this identity is imposed only when the 4 half-edges belong to distinct edges, but
+in fact this is redundant: if the 4 half-edges do not belong to distinct edges, then
+the identity already holds in Graphundec. So in fact imposing our relation IHmod
+is identical to imposing their relation IHbis
+0 .
+Theorem 5.7. Upon inverting χ − 2, χ − 3, and χ − 4, the maps
+Graphundec(S)
+(IHmod)
+⊗ det QS
+Φ
+−→ Graphtri(S)
+inc
+−→ Graph(S)
+are isomorphisms.
+Proof. Let R := Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1] and implicitly base change
+to this ring. We have already shown in Corollary 5.3 that the second map is an
+isomorphism, and Φ is certainly an epimorphism, so it remains to show that the
+composition is a monomorphism.
+
+28
+OSCAR RANDAL-WILLIAMS
+For an undecorated trivalent graph Γ, define a double edge to be an unordered
+pair of vertices which share precisely two edges, and a triple edge to be an unordered
+pair of vertices which share precisely three edges, i.e. form a theta-graph. Define
+µ(Γ) := 2 · #double edges of Γ + 3 · #triple edges of Γ,
+filter Graphundec by letting F kGraphundec be spanned by those Γ with µ(Γ) ≥ k,
+and give Graphundec/(IHmod) the induced filtration.
+If Γ = ΓH is a graph with µ(Γ) = k and a distinguished “H” subgraph, and ΓI
+is obtained by replacing this “H”-subgraph by “I”, then by applying the relation
+IHmod to this subgraph we find that
+(i) if the edge involved is not part of a double or triple edge then the relation
+gives ΓH − ΓI ∈ F k+1Graphundec/(IHmod),
+(ii) if the edge involved is part of a double or triple edge then the relation is trivial
+(i.e. already holds in Graphundec).
+Thus the associated graded of the induced filtration on Graphundec/(IHmod) can
+be described as Graphundec/(IH0), where as in [GN98] the relation IH0 means
+imposing the “I = H” relation when the four half-edges belong to different edges.
+Now IH0 is an equivalence relation on the set of isomorphism classes of trivalent
+graphs without loops, and similarly to [GN98, Proof of Proposition 2.3 (c)] it is
+easy to see that all connected trivalent graph without loops of the same rank and
+with the same legs are equivalent to each other: in other words the equivalences
+classes of such are given by partitions of S (the parts are the legs of each connected
+component) labelled by a power of e (recording the rank of the graph). It follows
+that the rank of Graphundec/(IHmod) in each degree, as an R-module, is at most that
+of Graph(∅) as determined in Lemma 5.1, and so the composition in the statement
+of the theorem, which is an epimorphism, must be an isomorphism.
+□
+5.5. On the work of Garoufalidis and Nakamura. The discussion of the last
+few sections can be used to complete the work of Garoufalidis and Nakamura [GN98,
+GN07], concerning the calculation of the invariants [Λ∗V13/(V22)]Sp in a stable
+range. Here we write Vλ for the irreducible Sp-representation corresponding to the
+partition λ, which was written as [λ]sp in those papers, and V22 denotes the unique
+copy of this irreducible in Λ2V13.
+Combining Theorem 1.1 and Proposition 2.3
+(c) of [GN98] was supposed to calculate [Λ∗V13/(V22)]Sp in a stable range, but for
+the corrected version of Theorem 1.1 in [GN07], which expresses these invariants as
+Graphundec(∅)g/(IHbis
+0 ), the authors say “it turns out that a simple stable structure
+of [these invariants] as in Proposition 2.3 (c) will not be easy to detect”. However
+Theorem 5.7 and equation (5.2) gives that
+[Λ∗V13/(V22)]Sp ∼= Graphundec(∅)g/(IHbis
+0 ) ∼= Graph(∅)g ∼= Q[Γ1, Γ2, . . .]
+in a stable range. Thus in fact Proposition 2.3 (c) of [GN98] is correct as stated.
+Remark 5.8. This can also be obtained from the work of Felder, Naef, and Willwacher
+[FNW21]. Specifically, the graded-commutative algebra A(g) defined just before
+Theorem 6 of that paper is Λ∗V13/(V22), and Theorem 6 together with Proposition
+36 (3) also gives the above.
+References
+[Aka05]
+H. Akazawa, Symplectic invariants arising from a Grassmann quotient and trivalent
+graphs, Math. J. Okayama Univ. 47 (2005), 99–117.
+[Bol12]
+S. K. Boldsen, Improved homological stability for the mapping class group with inte-
+gral or twisted coefficients, Math. Z. 270 (2012), no. 1-2, 297–329.
+[FNW21]
+M. Felder, F. Naef, and T. Willwacher, Stable cohomology of graph complexes,
+https://arxiv.org/abs/2106.12826, 2021.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+29
+[GN98]
+S. Garoufalidis and H. Nakamura, Some IHX-type relations on trivalent graphs and
+symplectic representation theory, Math. Res. Lett. 5 (1998), no. 3, 391–402.
+[GN07]
+, Corrigendum: “Some IHX-type relations on trivalent graphs and symplectic
+representation theory” [Math. Res. Lett. 5 (1998), no. 3, 391–402], Math. Res. Lett.
+14 (2007), no. 4, 689–690.
+[GRW14]
+S. Galatius and O. Randal-Williams, Stable moduli spaces of high-dimensional man-
+ifolds, Acta Math. 212 (2014), no. 2, 257–377.
+[GRW17]
+, Homological stability for moduli spaces of high dimensional manifolds. II,
+Ann. of Math. (2) 186 (2017), no. 1, 127–204.
+[GRW18]
+, Homological stability for moduli spaces of high dimensional manifolds. I, J.
+Amer. Math. Soc. 31 (2018), no. 1, 215–264.
+[GRW19]
+, Moduli spaces of manifolds: a user’s guide, Handbook of homotopy theory,
+Chapman & Hall/CRC, CRC Press, Boca Raton, FL, 2019, pp. 445–487.
+[GTMW09] S. Galatius, U. Tillmann, I. Madsen, and M. Weiss, The homotopy type of the cobor-
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+R. Hain, Johnson homomorphisms, EMS Surv. Math. Sci. 7 (2020), no. 1, 33–116.
+[Joh85]
+D. Johnson, The structure of the Torelli group. III. The abelianization of T , Topology
+24 (1985), no. 2, 127–144.
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+N. Kawazumi and S. Morita, The primary approximation to the cohomology of the
+moduli space of curves and cocycles for the stable characteristic classes, Math. Res.
+Lett. 3 (1996), no. 5, 629–641.
+[KM01]
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+primary
+approximation
+to
+the
+cohomology
+of
+the
+mod-
+uli
+space
+of
+curves
+and
+cocycles
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+Mumford-Morita-Miller
+classes,
+www.ms.u-tokyo.ac.jp/preprint/pdf/2001-13.pdf, 2001.
+[Kre79]
+M.
+Kreck,
+Isotopy
+classes
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+diffeomorphisms
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+(k − 1)-connected
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+parallelizable 2k-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ.
+Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979,
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+A. Kupers and O. Randal-Williams, The cohomology of Torelli groups is algebraic,
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+Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
+

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+arXiv:2301.04940v1  [math.AG]  12 Jan 2023
+FAILURE OF LEFSCHETZ HYPERPLANE THEOREM
+ANANYO DAN
+Abstract. In this article, we give a counterexample to the Lefschetz hyperplane theorem
+for non-singular quasi-projective varieties. A classical result of Hamm-Lˆe shows that Lefschetz
+hyperplane theorem can hold for hyperplanes in general position. We observe that the condition
+of “hyperplane” is strict in the sense that it is not possible to replace it by higher degree
+hypersurfaces. The counterexample is very simple: projective space minus finitely many points.
+Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails
+in the quasi-projective case.
+1. Introduction
+The underlying field will always be C.
+Consider a non-singular, projective variety Y of
+dimension n. The Lefschetz hyperplane theorem (LHT) states that for any hypersurface X ⊂ Y
+with OX(Y ) very ample, the restriction morphism
+Hk(Y, Z) → Hk(X, Z) is an isomorphism for all k < n − 1 and injective for k = n − 1.
+(1.1)
+If Y is the projective space, then the theorem extends further. In particular, the restriction from
+Hn−1(Pn) to Hn−1(X) is an isomorphism for a very general hypersurface X. The geometry of
+the locus of hypersurfaces where this isomorphism fails (also known as the Noether-Lefschetz
+locus), has been extensively studied [1–4, 11, 12].
+It is therefore evident that the failure of
+the Lefschetz hyperplane theorem can give rise to important questions in Hodge theory and
+deformation theory. The goal of this article is to investigate the failure of this theorem in the
+quasi-projective case.
+It was observed by Hamm and Lˆe [6, 7] that if a hyperplane section X in a quasi-projective
+variety Y is in “general” position, then (1.1) holds true. The criterion for general position, is
+given explicitly in terms of a Whitney stratification of Y (see §2.2). This leads to the natural
+question:
+Question: Is the Hamm-Lˆe theorem (Theorem 2.1) true if we replace “hyperplane” by higher
+degree hypersurface?
+This is true in the case when Y is a projective, non-singular variety. Surprisingly, this can
+fail even if Y is the complement of a single point in a projective space. In particular, we give
+an example of a higher degree hypersurface which satisfies all the conditions in the Hamm-Lˆe
+theorem except for being a hyperplane. Yet, in this case LHT fails. We now discuss this in
+details. Recall, a projective variety X is called non-factorial if the rank of the divisor class
+group Div(X) (i.e., the free abelian group of divisors on X modulo linear equivalence) is not
+the same as the rank of the Picard group Pic(X). We prove:
+Date: January 13, 2023.
+2020 Mathematics Subject Classification. 14C30, 32S35, 32S50.
+Key words and phrases. Hodge theory, Lefschetz hyperplane theorem, quasi-projective varieties, factoriality,
+Grothendieck-Lefschetz theorem, Picard group.
+1
+
+2
+ANANYO DAN
+Theorem 1.1. Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with
+n ≥ 4. Denote by Xsing the singular locus of X. Then, the natural restriction morphism
+H2(Pn\Xsing, Z) → H2(X\Xsing, Z)
+is not surjective.
+Using this theorem we now give an explicit example.
+Example 1.2. Let X ⊂ P4 be a hypersurface defined by the equation X2
+0 + X2
+1 + X2
+2 + X2
+3,
+where X0, ..., X4 are the coordinates on P4. Clearly, X has exactly one singular point x = [0 :
+0 : 0 : 0 : 1]. The divisor class group Div(X) is isomorphic to Z ⊕ Z (see [8, Ex. II.6.5]). By
+Lefschetz hyperplane theorem, we have H2(X, Z) ∼= Z. Using the exponential exact sequence,
+one can check that Pic(X) ∼= Z. Hence, X is non-factorial. Theorem 1.1 then implies that the
+restriction morphism from H2(P4\{x}, Z) to H2(X\{x}, Z) is not surjective.
+As an intermediate step we show that the Grothendieck-Lefschetz theorem [5] fails in the
+quasi-projective case (see Remark 3.2).
+Acknowledgement: I thank Dr. I. Kaur for discussions. The author was funded by EPSRC
+grant number EP/T019379/1.
+2. On the Hamm-Lˆe result
+In [7], Hamm and Lˆe proved a version of the Lefschetz hyperplane theorem for quasi-projective
+varieties (see Theorem 2.1 below). The proof follows in two stages. We use notations as in §2.1
+below. The first step is to check that for all i ≤ dim(Y ) − 2, Hi(Y \Z) (resp. Hm−1(Y \Z)) is
+isomorphic to (resp. contained in) the i-th (resp. (m − 1)-th) cohomology of Vr(L) ∩ (Y \Z),
+for some neighbourhood Vr(L) of L of “radius” r, for almost all r > 0 (see [7, Theorem 1.1.1]).
+The second step is to check whether L ∩ (Y \Z) is a deformation retract of Vr(L) ∩ (Y \Z). One
+observes that this holds true if L is in a “general” position. An explicit description of the general
+position will be mentioned in Theorem 2.1 below.
+2.1. Setup. Let Y be a projective subvariety of dimension m in Pn, Z ⊂ Y be an algebraic
+subspace and L ⊂ Pn a hyperplane in Pn such that Y \(Z ∪ L) is non-singular. Consider a
+stratification {Yi}i∈I of Y satisfying the following conditions:
+(1) each Yi is a real semi-algebraic subset of Y ,
+(2) {Yi} is a Whitney stratification,
+(3) Z is a union of some of the strata,
+(4) the stratification satisfies the Thom condition for the following function:
+τ : Y → R, sending y ∈ Y to
+k�
+i=1
+|fi(y)|2d/di
+n�
+i=0
+|yi|2d
+, where y = (y1, ..., yn),
+Z is defined by the homogeneous polynomials f1, ..., fk of degrees di, respectively and d
+is the l.c.m. of the di’s. See [10, §1.4.4] for the precise definition.
+2.2. On the Hamm-Lˆe result. Let Ω be the set of complex projective hyperplanes of Pn
+transverse to all the strata Yi.
+
+LEFSCHETZ THEOREM
+3
+Theorem 2.1. (Hamm-Lˆe [7, Theorem 1.1.3]) Assume that Y \Z is non-singular. Then, for any
+L ∈ Ω we have
+Hk(Y \Z, L ∩ (Y \Z)) = 0 for all k ≤ m − 1.
+In other words, the natural morphism from Hk(Y \Z, Z) to Hk(L∩(Y \Z), Z) is an isomorphism
+for all k ≤ m − 2 and injective for k = m − 1.
+We now write the stratification relevant to Example 1.2.
+Remark 2.2. Take Y = P4 ⊂ P5 defined by z5 = 0, where zi are the coordinates on P5. Take
+Z := [0, 0, 0, 0, 1, 0] the closed point in Y . Take the stratification of Y consisting of
+(Y \Z)
+�
+Z.
+Then, the equations defining Z in P5 are given by fi := zi for 0 ≤ i ≤ 3 and f5 := z5. The
+function τ is simply
+τ :=
+|z5|2 +
+3�
+i=0
+|zi|2
+5�
+i=0
+|zi|2
+.
+Note that this stratification satisfies conditions (1)-(4) in §2.1 above, with the stratification on R
+given by R\{0} �{0}. Finally, note that the hypersurface X in P5 defined by z2
+0 +z2
+1 +z2
+2 +z2
+3+z2
+5
+is singular at the point Z. As a result X is transverse to all the strata of Y . We will observe in
+Theorem 1.1 that if we replace L in Theorem 2.1 above by X, then the conclusion fails.
+3. Proof of Main theorem
+We will assume that the reader has basic familiarity with local cohomology. See [9] for basic
+definitions and results in this topic.
+Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with n ≥ 4. Denote by
+Xsing the singular locus of X, Y := Pn\Xsing and Xsm := X\Xsing. We first show:
+Proposition 3.1. The cohomology groups H1(OY ), H2(OY ) and H1(OXsm) all vanish, in both
+analytic as well as Zariski topology.
+Proof. Recall, the long exact sequence for local cohomology groups, which exists in both topolo-
+gies (see [9, Corollary 1.9]):
+... → H1(OPn) → H1(OY ) → H2
+Xsing(OPn) → H2(OPn) → H2(OY ) → H3
+Xsing(OPn) → ...
+Recall, H1(OPn) = 0 = H2(OPn). By Serre’s GAGA, H1(O
+an
+Pn) = 0 = H2(O
+an
+Pn). To prove the
+vanishing of H1(OY ) and H2(OY ), we simply need to prove the vanishing of Hi
+Xsing(OPn) for
+i = 2, 3 in both topologies.
+Consider the spectral sequence (see [9, Proposition 1.4]):
+Ep,q
+2
+= Hp(Pn, Hq
+Xsing(OPn)) ⇒ Hp+q
+Xsing(OPn).
+(3.1)
+We are interested in the cases when p + q equals 2 or 3. Since n ≥ 4 and Xsing are closed points,
+we have (see [13, Proposition 1.2])
+Hq
+Xsing(OPn) = 0 for q ≤ 3.
+This implies that Ep,q
+2
+= 0 for p + q equals 2 or 3. Hence the spectral sequence degenerates at
+E2 in this case and Hi
+Xsing(OPn) = 0 in both topologies. This proves the vanishing of H1(OY )
+and H2(OY ).
+
+4
+ANANYO DAN
+The proof for the vanishing of H1(OXsm) follows similarly. In particular, using [9, Corollary
+1.9], it suffices to check the vanishing of H1(OX) and H2
+Xsing(OX). Since X is a hypersurface
+in Pn and n ≥ 4, H1(OX) = 0.
+By Serre’s GAGA, H1(O
+an
+X ) = 0. To prove the vanishing
+of H2
+sing(OX) use the spectral sequence (3.1) above after replacing Pn by X and p + q = 2.
+Since dim X ≥ 3, [13, Proposition 1.2] implies that Hq
+Xsing(OX) = 0 for q ≤ 2. This implies
+that the spectral sequence degenerates at E2 and H2
+Xsing(OX) = 0 in both topologies. Hence,
+H1(OXsm) = 0 in both topologies. This proves the proposition.
+□
+Proof of the main theorem. We prove the theorem by contradiction. Suppose that the restric-
+tion morphism from H2(Y, Z) to H2(X, Z) is surjective. Comparing the long exact sequences
+associated to the exponential exact sequence for Y and Xsm we get the following diagram where
+the horizontal rows are exact:
+H1(OY )
+✲ H1(O∗
+Y )
+∂1✲ H2(Y, Z)
+✲ H2(OY )
+⟲
+⟲
+⟲
+H1(OXsm)
+❄
+✲ H1(O∗
+Xsm)
+ρ′
+❄
+∂2✲ H2(Xsm, Z)
+ρ
+❄
+✲ H2(OXsm)
+❄
+(3.2)
+Using the vanishing results from Proposition 3.1, we conclude that ∂1 is an isomorphism and ∂2
+is injective. By assumption, ρ is surjective. We claim that ρ′ is surjective. Indeed, given α ∈
+H1(O∗
+Xsm), the surjectivity of ρ implies that there exists β ∈ H2(Y, Z) such that ρ(β) = ∂2(α).
+Since ∂1 is an isomorphism, there exist α′ ∈ H1(O∗
+Y ) mapping to β via ∂1. Using the injectivity
+of ∂2 and the commutativity of the middle square, we have ρ′(α′) = α. This proves the claim.
+Since ρ′ is surjective, we have the following surjective morphism:
+Z = Pic(Pn) ∼= Pic(Y )
+ρ′
+։ Pic(Xsm) ∼= Div(X)
+(3.3)
+where the second and the last isomorphisms follow from the fact that Xsing is of codimensional
+at least 2 in X and Pn. By Lefschetz hyperplane theorem, we have H2(X, Z) ∼= H2(Pn, Z) = Z,
+generated by the class of the hyperplane section. Note that, H1(OX) and H2(OX) vanish (use [8,
+Ex. III.5.5] and n ≥ 4). Using the exponential short exact sequence for X, we conclude that
+Pic(X) ∼= Z. Combining with (3.3), this implies rk Div(X) = rk Pic(X). But this contradicts
+the fact that X is non-factorial. Hence, the restriction morphism from H2(Y, Z) to H2(X, Z)
+cannot be surjective. This proves the theorem.
+□
+Remark 3.2. Let X be as in Theorem 1.1. Then, the restriction morphism
+Pic(Pn\Xsing) → Pic(X\Xsing)
+is not surjective. Indeed,
+Pic(Pn\Xsing) ∼= Pic(Pn) ∼= Z and Pic(X\Xsing) ∼= Div(X).
+By Lefschetz hyperplane theorem for projective hypersurfaces, we have Pic(X) ∼= Z. Since X is
+non-factorial, the rank of Div(X) is not the same as that of Pic(X). Therefore, Pic(Pn\Xsing)
+cannot be isomorphic to Pic(X\Xsing).
+References
+[1] C. Ciliberto, J. Harris, and R. Miranda. General components of the Noether-Lefschetz locus and their density
+in the space of all surfaces. Mathematische Annalen, 282(4):667–680, 1988.
+[2] A. Dan. On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci. Communications
+in Contemporary Mathematics, 17(5):1550002, 2015.
+[3] A. Dan. On a conjecture of Harris. Communications in Contemporary Mathematics, 23(07):2050028, 2021.
+
+LEFSCHETZ THEOREM
+5
+[4] M. Green. A new proof of the explicit Noether-Lefschetz theorem. J. Differential Geometry, 27:155–159, 1988.
+[5] A. Grothendieck. SGA 2. S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie-1962-Cohomologie locale des
+faisceaux coh´erents et th´eoremes de Lefschetz locaux et globaux (North-Holland, Amsterdam), 1968.
+[6] H. Hamm. Lefschetz theorems for singular varieties. In Proceedings of symposia in pure mathematics, vol-
+ume 40, pages 547–557. AMS, 1983.
+[7] H. Hamm and D. T. Lˆe. Lefschetz theorems on quasi-projective varieties. Bulletin de la Soci´et´e math´ematique
+de France, 113:123–142, 1985.
+[8] R. Hartshorne. Algebraic Geometry. Graduate text in Mathematics-52. Springer-Verlag, 1977.
+[9] R. Hartshorne. Local Cohomology: A Seminar Given by A. Groethendieck, Harvard University. Fall, 1961,
+volume 41. Springer, 2006.
+[10] D. T. Lˆe and B. Teissier. Cycles ´evanescents, sections planes et conditions de whitney ii, singularities, part
+2 (arcata, calif., 1981), 65-103. In Proc. Sympos. Pure Math, volume 40.
+[11] C. Voisin. Une pr´ecision concernant le th´eor`eme de Noether. Math. Ann., 280(4):605–611, 1988.
+[12] C. Voisin. Sur le lieu de Noether-Lefschetz en degr´es 6 et 7. Compositio Mathematica, 75(1):47–68, 1990.
+[13] Y. Yoshino. Maximal Cohen-Macaulay Modules Over Cohen-Macaulay Rings, volume 146. Cambridge Uni-
+versity Press, 1990.
+School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road,
+S3 7RH, UK
+Email address: [email protected]
+

TdE4T4oBgHgl3EQfLwwK/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='04940v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='AG]  12 Jan 2023  FAILURE OF LEFSCHETZ HYPERPLANE THEOREM  ANANYO DAN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In this article, we give a counterexample to the Lefschetz hyperplane theorem  for non-singular quasi-projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' A classical result of Hamm-Lˆe shows that Lefschetz  hyperplane theorem can hold for hyperplanes in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We observe that the condition  of “hyperplane” is strict in the sense that it is not possible to replace it by higher degree  hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The counterexample is very simple: projective space minus finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails  in the quasi-projective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Introduction  The underlying field will always be C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Consider a non-singular, projective variety Y of  dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The Lefschetz hyperplane theorem (LHT) states that for any hypersurface X ⊂ Y  with OX(Y ) very ample, the restriction morphism  Hk(Y, Z) → Hk(X, Z) is an isomorphism for all k < n − 1 and injective for k = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1)  If Y is the projective space, then the theorem extends further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In particular, the restriction from  Hn−1(Pn) to Hn−1(X) is an isomorphism for a very general hypersurface X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The geometry of  the locus of hypersurfaces where this isomorphism fails (also known as the Noether-Lefschetz  locus), has been extensively studied [1–4, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  It is therefore evident that the failure of  the Lefschetz hyperplane theorem can give rise to important questions in Hodge theory and  deformation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The goal of this article is to investigate the failure of this theorem in the  quasi-projective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  It was observed by Hamm and Lˆe [6, 7] that if a hyperplane section X in a quasi-projective  variety Y is in “general” position, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The criterion for general position, is  given explicitly in terms of a Whitney stratification of Y (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This leads to the natural  question:  Question: Is the Hamm-Lˆe theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1) true if we replace “hyperplane” by higher  degree hypersurface?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  This is true in the case when Y is a projective, non-singular variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Surprisingly, this can  fail even if Y is the complement of a single point in a projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In particular, we give  an example of a higher degree hypersurface which satisfies all the conditions in the Hamm-Lˆe  theorem except for being a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Yet, in this case LHT fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We now discuss this in  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Recall, a projective variety X is called non-factorial if the rank of the divisor class  group Div(X) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', the free abelian group of divisors on X modulo linear equivalence) is not  the same as the rank of the Picard group Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We prove:  Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' 14C30, 32S35, 32S50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hodge theory, Lefschetz hyperplane theorem, quasi-projective varieties, factoriality,  Grothendieck-Lefschetz theorem, Picard group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  1  2  ANANYO DAN  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with  n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Denote by Xsing the singular locus of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Then, the natural restriction morphism  H2(Pn\\Xsing, Z) → H2(X\\Xsing, Z)  is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Using this theorem we now give an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let X ⊂ P4 be a hypersurface defined by the equation X2  0 + X2  1 + X2  2 + X2  3,  where X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', X4 are the coordinates on P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Clearly, X has exactly one singular point x = [0 :  0 : 0 : 0 : 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The divisor class group Div(X) is isomorphic to Z ⊕ Z (see [8, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By  Lefschetz hyperplane theorem, we have H2(X, Z) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Using the exponential exact sequence,  one can check that Pic(X) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence, X is non-factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 then implies that the  restriction morphism from H2(P4\\{x}, Z) to H2(X\\{x}, Z) is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  As an intermediate step we show that the Grothendieck-Lefschetz theorem [5] fails in the  quasi-projective case (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Acknowledgement: I thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Kaur for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The author was funded by EPSRC  grant number EP/T019379/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On the Hamm-Lˆe result  In [7], Hamm and Lˆe proved a version of the Lefschetz hyperplane theorem for quasi-projective  varieties (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The proof follows in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We use notations as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The first step is to check that for all i ≤ dim(Y ) − 2, Hi(Y \\Z) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hm−1(Y \\Z)) is  isomorphic to (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' contained in) the i-th (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' (m − 1)-th) cohomology of Vr(L) ∩ (Y \\Z),  for some neighbourhood Vr(L) of L of “radius” r, for almost all r > 0 (see [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  The second step is to check whether L ∩ (Y \\Z) is a deformation retract of Vr(L) ∩ (Y \\Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' One  observes that this holds true if L is in a “general” position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' An explicit description of the general  position will be mentioned in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let Y be a projective subvariety of dimension m in Pn, Z ⊂ Y be an algebraic  subspace and L ⊂ Pn a hyperplane in Pn such that Y \\(Z ∪ L) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Consider a  stratification {Yi}i∈I of Y satisfying the following conditions:  (1) each Yi is a real semi-algebraic subset of Y ,  (2) {Yi} is a Whitney stratification,  (3) Z is a union of some of the strata,  (4) the stratification satisfies the Thom condition for the following function:  τ : Y → R, sending y ∈ Y to  k�  i=1  |fi(y)|2d/di  n�  i=0  |yi|2d  , where y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', yn),  Z is defined by the homogeneous polynomials f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', fk of degrees di, respectively and d  is the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' of the di’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' See [10, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='4] for the precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On the Hamm-Lˆe result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let Ω be the set of complex projective hyperplanes of Pn  transverse to all the strata Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  LEFSCHETZ THEOREM  3  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' (Hamm-Lˆe [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='3]) Assume that Y \\Z is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Then, for any  L ∈ Ω we have  Hk(Y \\Z, L ∩ (Y \\Z)) = 0 for all k ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  In other words, the natural morphism from Hk(Y \\Z, Z) to Hk(L∩(Y \\Z), Z) is an isomorphism  for all k ≤ m − 2 and injective for k = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  We now write the stratification relevant to Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Take Y = P4 ⊂ P5 defined by z5 = 0, where zi are the coordinates on P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Take  Z := [0, 0, 0, 0, 1, 0] the closed point in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Take the stratification of Y consisting of  (Y \\Z)  �  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Then, the equations defining Z in P5 are given by fi := zi for 0 ≤ i ≤ 3 and f5 := z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The  function τ is simply  τ :=  |z5|2 +  3�  i=0  |zi|2  5�  i=0  |zi|2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Note that this stratification satisfies conditions (1)-(4) in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 above, with the stratification on R  given by R\\{0} �{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Finally, note that the hypersurface X in P5 defined by z2  0 +z2  1 +z2  2 +z2  3+z2  5  is singular at the point Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' As a result X is transverse to all the strata of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We will observe in  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 that if we replace L in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 above by X, then the conclusion fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Proof of Main theorem  We will assume that the reader has basic familiarity with local cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' See [9] for basic  definitions and results in this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Denote by  Xsing the singular locus of X, Y := Pn\\Xsing and Xsm := X\\Xsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We first show:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The cohomology groups H1(OY ), H2(OY ) and H1(OXsm) all vanish, in both  analytic as well as Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Recall, the long exact sequence for local cohomology groups, which exists in both topolo-  gies (see [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='9]):  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' → H1(OPn) → H1(OY ) → H2  Xsing(OPn) → H2(OPn) → H2(OY ) → H3  Xsing(OPn) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Recall, H1(OPn) = 0 = H2(OPn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By Serre’s GAGA, H1(O  an  Pn) = 0 = H2(O  an  Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' To prove the  vanishing of H1(OY ) and H2(OY ), we simply need to prove the vanishing of Hi  Xsing(OPn) for  i = 2, 3 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Consider the spectral sequence (see [9, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='4]):  Ep,q  2  = Hp(Pn, Hq  Xsing(OPn)) ⇒ Hp+q  Xsing(OPn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1)  We are interested in the cases when p + q equals 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Since n ≥ 4 and Xsing are closed points,  we have (see [13, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2])  Hq  Xsing(OPn) = 0 for q ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  This implies that Ep,q  2  = 0 for p + q equals 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence the spectral sequence degenerates at  E2 in this case and Hi  Xsing(OPn) = 0 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the vanishing of H1(OY )  and H2(OY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  4  ANANYO DAN  The proof for the vanishing of H1(OXsm) follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In particular, using [9, Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='9], it suffices to check the vanishing of H1(OX) and H2  Xsing(OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Since X is a hypersurface  in Pn and n ≥ 4, H1(OX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  By Serre’s GAGA, H1(O  an  X ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' To prove the vanishing  of H2  sing(OX) use the spectral sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1) above after replacing Pn by X and p + q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Since dim X ≥ 3, [13, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2] implies that Hq  Xsing(OX) = 0 for q ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This implies  that the spectral sequence degenerates at E2 and H2  Xsing(OX) = 0 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence,  H1(OXsm) = 0 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  □  Proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We prove the theorem by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Suppose that the restric-  tion morphism from H2(Y, Z) to H2(X, Z) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Comparing the long exact sequences  associated to the exponential exact sequence for Y and Xsm we get the following diagram where  the horizontal rows are exact:  H1(OY )  ✲ H1(O∗  Y )  ∂1✲ H2(Y, Z)  ✲ H2(OY )  ⟲  ⟲  ⟲  H1(OXsm)  ❄  ✲ H1(O∗  Xsm)  ρ′  ❄  ∂2✲ H2(Xsm, Z)  ρ  ❄  ✲ H2(OXsm)  ❄  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2)  Using the vanishing results from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1, we conclude that ∂1 is an isomorphism and ∂2  is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By assumption, ρ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We claim that ρ′ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Indeed, given α ∈  H1(O∗  Xsm), the surjectivity of ρ implies that there exists β ∈ H2(Y, Z) such that ρ(β) = ∂2(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Since ∂1 is an isomorphism, there exist α′ ∈ H1(O∗  Y ) mapping to β via ∂1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Using the injectivity  of ∂2 and the commutativity of the middle square, we have ρ′(α′) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Since ρ′ is surjective, we have the following surjective morphism:  Z = Pic(Pn) ∼= Pic(Y )  ρ′  ։ Pic(Xsm) ∼= Div(X)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='3)  where the second and the last isomorphisms follow from the fact that Xsing is of codimensional  at least 2 in X and Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By Lefschetz hyperplane theorem, we have H2(X, Z) ∼= H2(Pn, Z) = Z,  generated by the class of the hyperplane section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Note that, H1(OX) and H2(OX) vanish (use [8,  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='5] and n ≥ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Using the exponential short exact sequence for X, we conclude that  Pic(X) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Combining with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='3), this implies rk Div(X) = rk Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' But this contradicts  the fact that X is non-factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence, the restriction morphism from H2(Y, Z) to H2(X, Z)  cannot be surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let X be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Then, the restriction morphism  Pic(Pn\\Xsing) → Pic(X\\Xsing)  is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Indeed,  Pic(Pn\\Xsing) ∼= Pic(Pn) ∼= Z and Pic(X\\Xsing) ∼= Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  By Lefschetz hyperplane theorem for projective hypersurfaces, we have Pic(X) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Since X is  non-factorial, the rank of Div(X) is not the same as that of Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Therefore, Pic(Pn\\Xsing)  cannot be isomorphic to Pic(X\\Xsing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Ciliberto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Harris, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Miranda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' General components of the Noether-Lefschetz locus and their density  in the space of all surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Mathematische Annalen, 282(4):667–680, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Communications  in Contemporary Mathematics, 17(5):1550002, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On a conjecture of Harris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Communications in Contemporary Mathematics, 23(07):2050028, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  LEFSCHETZ THEOREM  5  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' A new proof of the explicit Noether-Lefschetz theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Differential Geometry, 27:155–159, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' SGA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie-1962-Cohomologie locale des  faisceaux coh´erents et th´eoremes de Lefschetz locaux et globaux (North-Holland, Amsterdam), 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hamm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lefschetz theorems for singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In Proceedings of symposia in pure mathematics, vol-  ume 40, pages 547–557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' AMS, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hamm and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lˆe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lefschetz theorems on quasi-projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Bulletin de la Soci´et´e math´ematique  de France, 113:123–142, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Algebraic Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Graduate text in Mathematics-52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Springer-Verlag, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Local Cohomology: A Seminar Given by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Groethendieck, Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Fall, 1961,  volume 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lˆe and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Teissier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Cycles ´evanescents, sections planes et conditions de whitney ii, singularities, part  2 (arcata, calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', 1981), 65-103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Pure Math, volume 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Voisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Une pr´ecision concernant le th´eor`eme de Noether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', 280(4):605–611, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Voisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Sur le lieu de Noether-Lefschetz en degr´es 6 et 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Compositio Mathematica, 75(1):47–68, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Yoshino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Maximal Cohen-Macaulay Modules Over Cohen-Macaulay Rings, volume 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Cambridge Uni-  versity Press, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road,  S3 7RH, UK  Email address: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='dan@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='uk' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}

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@@ -0,0 +1,3030 @@
+Dynamics and stability of the two-body problem with Yukawa
+correction to Newton’s gravity, revisited and applied
+numerically to the solar system
+Nawras Abo Hasan1∗, Nabil Joudieh1†and Nidal Chamoun2‡
+1 Physics Department, Damascus University, Damascus, Syria
+2 Physics Department, HIAST, P.O. Box 31983, Damascus, Syria
+Abstract
+In this manuscript, we review the motion of two-body celestial system (planet-sun) for
+a Yukawa-type correction on Newton’s gravitational potential using Hamilton’s formulation.
+We reexamine the stability using the corresponding linearization Jacobian matrix, and verify
+that the Bertrand’s theorem conditions are met for radii ≪ 1015m, and so bound closed orbits
+are expected. Applied to the solar system, we present the equation of motion of the planet,
+then solve it both analytically and numerically. Making use of the analytical expression of
+the orbit, we estimate the Yukawa strength α, and find it larger than the nominal value
+(10−8) adopted in previous studies, in that it is of order (α = 10−4 − 10−5) for terrestrial
+planets (Mercury, Venus, earth, Mars and Pluto) whereas it is even larger (α = 10−3) for
+the Giant planets (Jupiter, Saturn, Uranus and Neptune). Taking as inputs (rmin, vmas, e)
+observed by NASA, we analyse the orbits analytically and numerically for both the estimated
+and nominal values of α, and determine the corresponding trajectories. For each obtained
+orbit we recalculate the characterizing parameters (rmin, rmax, a, b, e) and compare their
+values according to the used potential (Newton with/without Yukawa correction) and to the
+method used (analytical and/or numerical). When compared to the observational data, we
+conclude that the correction on the path due to Yukawa correction is of order of and up to
+80 million km (20 million km) as a maximum deviation occurring for Neptune (Pluto) for
+nominal (estimated) value of α.
+Keywords: gravitational two-body problem, Yukawa potential, closed orbit
+’
+0
+Introduction
+The past several years have witnessed a resurgence of interest in experimental testing of gravity,
+particularly in the possibility of deviations from the predictions of Newtonian gravity, which is
+considered as an excellent approximation of General Relativity (GR) on large distance scale [1].
+Many theoretical models suggest the existence of new, relatively weak, intermediate-range force
+coexisting with gravity such that the net resulting interaction would behave like a new correction
+∗[email protected]
+†[email protected]
+‡[email protected]
+1
+arXiv:2301.02498v1  [astro-ph.EP]  6 Jan 2023
+
+to the potentials defining the gravitational field. It is known [2] that there are only two types
+of central potentials, namely the Newton 1r and the Harmonic r2 potentials, where ANY finite
+motion of an object, subject to this central potential, leads to a closed path (Bertrands theorem).
+There are some ‘exceptions’ to this statement, in the sense that there might be closed bound
+trajectories for a central potential different from the Newton and Harmonic ones, which have
+been studied in [3, 4]. In this contribution, we revisit the effect of a Yukawa correction to the
+gravitational force over large distances.
+Theories of massive gravity [5,6], adding a mass term to the graviton (the carrier of gravity),
+have raised a wide interest and the Yukawa potential is the popular parametrization of such
+theories. Actually, many works describing deviations from Newtons inverse square law have
+addressed the Yukawa-type correction. Assuming gravity is exerted by exchanges of gravitons,
+it is clear that a test for the graviton mass (µg) is to ask whether the Newton (1/r) potential
+shows any evidence of dying at large distances because of Yukawa exponential cutoff (e−µgr).
+Since the seventies [7], bounds on the gravitons mass (µg ≤ 1.1 × 10−29 eV) were used to
+put a bound on its compton wavelength considered as a distance scale for Yukawa correction
+(λ ∼ 2π
+µg ≥ 3.7 MPc). The authors of [8] gave a bound on the Yukawa range (λ) in the order of
+(101 − 104 AU) corresponding to (µg ≤ 10−24) eV.
+Theories like Scalar-Tensor-Vector Gravity Theory [9] predict a Yukawa-like fifth force. The
+authors of [10], showed that screened modified gravity can suppress the fifth force in dense regions
+and allow theories to evade the solar system and laboratory tests of the weak equivalence princi-
+ple. In [11], an extended theory of gravity, with a modified potential including post-Newtonian
+terms, whose expansion is different from that of Yukawa correction, called ‘vacuum bootstrapped
+Newtonian gravity’, was subjected to solar system tests, through a procedure which was applied
+to Yukawa corrections at the Galactic center [12], with no significant deviations from GR found.
+In [13], a Keplerian-type parametrization was shown as a solution of the equations of motion
+for a Yukawa-type potential between two bodies. In fact, the two-body solution for alternative
+theories yield a strong constraint for solar system [14, 15], whereas several analyses of Yukawa
+potential for a 2-body system in different contexts were carried out [16, 17]. The orbit of a
+single particle moving under Yukawa potential was studied in [18], and the precessing ellipse
+type orbits were observed. In [19], it was noted that the modified gravity with Yukawa-like
+long-range potential was (un)successful on astrophysical scales (in solar system), whereas an
+analysis of Yukawa potential in f(R) gravity was given in [20]. The work of [21] showed that
+a Yukawa fifth force is expected to be sub-dominant in satellite dynamics and space geodesy
+experiments, as long as they are performed at altitudes greater than a few hundred kilometres.
+The Yukawa strength was estimated in [22] to be (α < 10−5-10−8) for distances of order 109
+cm, whereas the use of laser data from LAGEOS satellites yield a constraint on α of the order
+of 10−12.
+In this letter, we build on work from [23], in which the dynamics and stability of the two body
+problem with a Newtonian potential corrected by a Yukawa term were explored. In particular,
+we reproduced their analytical results and applied them to the study of all the planets of our solar
+system. Solving analytically the planet equation of motion, one finds an elliptical trajectory,
+which one can also obtain numerically using Runge-Kutta method. Starting from the observed
+values of the perihelion distance and velocity (rmin, vmax) and of the tranjectory eccentricity e,
+stated in NASA public results [24]∗, one could determine the ellipsis equation and estimate, for
+∗Although the standard deviations of the planetary trajectories are not quoted in the NASA public website,
+however one can consider that the corresponding error equals to the last digit of the quoted significant numbers.
+2
+
+Yukawa corrected potential, the Yukawa strength α. One can use this estimated value, or another
+nominal value taken from other studies, to either draw the analytical trajectory and recalculate
+the characterizing parameters: the shortest (longest) distance to the Sun rmin(rmax), the semi
+major (minor) axis a(b) and the eccentricity e, or to solve numerically the equations of motion
+with the Yukawa-corrected potential in order to check the closedness of the resulting trajectory,
+whose characteristics are to be reevaluated again. Later, we compared these results with those
+calculated for the Keplerian motion of planets subject to the pure Newtonian potential, and, in
+addition, showed the compatibility of the results with the observational NASA data.
+More specifically, for the two-body system (planet-sun), the Newtonian potential is given by:
+VN(r) = −GmpM⊙
+r
+(1)
+where G = 6.674×10−11 Nm2
+Kg2 is the gravitational Newton constant, mp (M⊙) is the planet (sun)
+mass. With a Yukawa correction, the gravitational potential becomes
+V (r) = −GmpM⊙
+r
+�
+1 + αe− r
+λ
+�
+= VN(r) + VY k(r)
+(2)
+where VY k is the Yukawa correction to the Newtonian potential and α (λ) represents the
+strength (range) of the Yukawa correction. Previous studies [23, 25] gave the nominal values
+(α = 10−8(λ = 103AU = 1015m). However, our estimations gave a larger order of magnitude for
+the Yukawa strength: α ∼ 10−4 − 10−5 for terrestrial planets (Mercury, Venus, Earth, Mars and
+Pluto) and α ∼ 10−3 for the remaining Giant planets (Jupiter, Saturn, Uranus and Neptune),
+which are in line with [13].
+We saw that for estimated α, the maximum deviation from observed data, which increases
+the further the planet is (20 million km in Pluto), is less than that of the α nominal value
+(80 million km in Neptune), which is plausible considering that the estimation of α is done by
+identifying the factor containing it to observational data.
+For each of the nominal and estimated values of α, we analysed the planet’s trajectory
+both analytically and numerically. Analytics wise, we started from the observational data of
+NASA (rmin, vmax, e) and reconstructed the closed ellipse trajectory of which we re-evaluated
+the characteristics (rmin, rmax, a, b, e) and compared with the pure Newton case and with the
+observational data. Numerics wise, the α determines the potential under which the planet moves,
+and so one can solve the equations of motion numerically using Runge-Kutta method taking as
+initial conditions the observed data of (rmin, vmax), to check that one gets closed trajectories in
+excellent agreement with the elliptical shapes, of which we can evaluate the characteristics that
+one compares to the pure Newtonian case, to the analytical method results and to the observed
+data.
+The manuscript is organized as follows.
+In section (1), we revise the system dynamics
+using Hamilton’s method. In section(2), we state the types of stability and determine the one
+corresponding to the system under study. We discuss, in section(3) and following [23], Bertrand’s
+theorem and get the analytical solution to the equation of motion. Finally, we apply in section
+(4) the obtained approximative analytical results to the study of the solar system planets in
+order to estimate the Yukawa strength and re-determine the trajectory characteristics for both
+estimated and nominal values of α, as well as solve numerically the equations. The results,
+of comparing the analytical/numerical outputs with the observed data according to the used
+In our computations, we used the whole digits allowed by machine precision, however the results in the appendices
+tables showed only significant digits equal to those of the observed data.
+3
+
+potential, are presented in form of plots for all the planets, whereas the corresponding tables
+are given in an appendix. We end up with conclusions in section (5).
+1
+Hamiltonian formulation
+We start with the Hamiltonian H = T + V where T is the kinetic energy of both masses and V
+is the Gravitational potential energy.
+H =
+⃗p2
+1
+2mp
++
+⃗p2
+2
+2M⊙
+−
+K
+|⃗r2 − ⃗r1|
+�
+1 + αe− |⃗r2−⃗r1|
+λ
+�
+(3)
+where ⃗ri, (⃗vi), i = 1, 2 are the positions (velocities) of the two masses with corresponding mo-
+menta p1 = M⊙v1, p2 = mpv2, K = GmpM⊙. Changing to the center of mass frame (c.o.m),
+with
+⃗r1 = +
+mp
+mp + M⊙
+⃗r = + µ
+M⊙
+⃗r + ⃗R
+,
+⃗r2 = −
+M⊙
+mp + M⊙
+⃗r = − µ
+mp
+⃗r + ⃗R
+(4)
+⃗r = ⃗r1 − ⃗r2
+,
+⃗R = M⊙⃗r1 + mp⃗r2
+mp + M⊙
+(5)
+⃗v1 = ˙⃗R +
+µ
+M⊙
+⃗v
+,
+⃗v2 = ˙⃗R + −µ
+mp
+⃗v
+(6)
+⃗v = ˙⃗r
+,
+⃗p = µ⃗v,
+(7)
+¨⃗R = ⃗0
+,
+µ¨⃗r = M⊙¨⃗r1 = −mp¨⃗r2,
+(8)
+we get
+H = 1
+2(M⊙ + mp) ˙⃗R2 + H
+:
+H = p2
+2µ − K
+r
+�
+1 + αe− r
+λ
+�
+(9)
+Here we have defined µ =
+mpM⊙
+mp+M⊙ as the reduced mass of the system and r = |⃗r|. We switch to
+polar coordinates in the c.o.m to get
+H = 1
+2µ
+�
+p2
+r + p2
+ϕ
+r2
+�
+− K
+r
+�
+1 + αe− r
+λ
+�
+(10)
+From the canonical equations ( [26]): ˙qi =
+�
+∂H
+∂pi
+�
+, ˙pi = −
+�
+∂H
+∂qi
+�
+, and since the Hamiltonian is
+cyclic in ϕ (i.e. it does not depend explicitly on ϕ), we have:
+˙ϕ = ∂H
+∂pϕ
+= pϕ
+µr2
+(11)
+˙pϕ = −∂H
+∂ϕ = 0 ⇒ pϕ = µr2 ˙ϕ = ℓ = constant
+(12)
+where ℓ is the angular momentum of the two-body system, and therefore Hamiltons equations
+for r become:
+˙r = ∂H
+∂pr
+= pr
+µ
+(13)
+˙pr = −∂H
+∂r = ℓ2
+µr3 − K
+r2
+�
+1 + α
+�
+1 + r
+λ
+�
+e− r
+λ
+�
+(14)
+4
+
+Figure 1:
+The reduced potential (red line) given for fixed angular momentum (Eq. 16). The
+pink line denotes the magnitude of the purely Yukawa term (
+���− αK
+r e− r
+λ
+���), whereas the blue line
+represents the Keplerian reduced potential, i.e. Eq. 16 without the Yukawa term.
+Again, and since H(t) = H(t0) = h is constant during the motion of the masses [26], and since
+p2
+r = µ2 ˙r2 ≥ 0 we get a lower bound for the total energy of the system:
+h ≥
+ℓ2
+2µr2 − K
+r
+�
+1 + αe− r
+λ
+�
+(15)
+The right hand side of eq. (15) is defined to be the “reduced potential”, which is common in the
+Kepler problem moving from two degrees of freedom to only one (with the Yukawa correction)
+Vred(r) =
+ℓ2
+2µr2 − K
+r
+�
+1 + αe− r
+λ
+�
+(16)
+One can draw the function for fixed ℓ giving the allowed regions of motion (look at figure 1).
+Note that µ > 0, λ > 0 and α > 0.
+2
+The linearization matrix
+Following [27], in order to determine the stability of the equilibrium points of the system, we
+must form a matrix differential equation using the system equations of motion (Hamiltons Eqs.
+13 and 14 for r, p). The linear system has the form:
+d
+dt
+�
+r
+pr
+�
+=
+�
+f(r, p)
+g(r, p)
+�
+=
+�
+f0
+g0
+�
+eq
++
+� ∂f
+∂r
+∂f
+∂pr
+∂g
+∂r
+∂g
+∂pr
+� �
+r
+pr
+�
+where f (r, pr) = pr
+µ ,
+g (r, pr) = ℓ2
+µr3 − K
+r2
+�
+1 + α
+�
+1 + r
+λ
+�
+e− r
+λ
+�
+(17)
+Given that λ = 1015m for orbits of size comparable to the solar system dimensions [28], one can
+assume that r
+λ is small enough that one can Taylor expand the exponential and ignore terms of
+5
+
+70
+Vred
+09
+Vkep
+-Vyuk
+50
+40
+>
+30
+20
+10
+0
+-10�
+r2
+λ2
+�
+, leading to:
+e− r
+λ ≈ 1 − r
+λ + O
+� r2
+λ2
+�
+≈ 1 − r
+λ
+(18)
+Thus
+g (r, pr) = ℓ2
+µr3 − K
+r2
+�
+1 + α
+�
+1 + r
+λ
+� �
+1 − r
+λ
+��
+≈ ℓ2
+µr3 − K
+r2 (1 + α),
+with the Yukawa effect within this approximation being limited to replacing K by K(1+α), which
+tells that the potential shape is still Newtonian (1/r), and according to Bertrand’s theorem every
+bound trajectory is thus closed for small r/λ. One can see this fact directly from Eq. (16) as
+it gives, compared to the Keplerian potential, within the approximation just a shift, in addition
+to the replacement (K → K(1 + α)), which does not interfere in the equations of motion:
+Vred(r)
+≈
+ℓ2
+2µr2 − K
+r (1 + α) + Kα
+λ .
+(19)
+Consequently, the Jacobian matrix takes the form:
+�
+˙r
+˙pr
+��
+=
+�
+0
+1
+µ
+−3ℓ2
+µr4 + 2K
+r3 (1 + α)
+0
+� �
+r
+pr
+�
+(20)
+where terms of order O
+�
+r2
+λ2
+�
+were ignored, and where the equilibrium point (r, pr)eq satisfies
+feq(r, pr) = geq(r, pr) = 0. We can determine the r at equilibrium using (eq. 14) to get upto
+leading order:
+req =
+ℓ2
+µK(1 + α)
+(21)
+We can now test for stability by choosing values of (α, µ, K, ℓ, λ) and finding the eigenvalues
+of the Jacobian matrix (20) after substituting the equilibrium solution found above (eq. 21).
+Recall that the eigenvalues β1, β2 are found by solving the following equation:
+det |J − βI2×2| = 0
+(22)
+with I2×2 referring to the 2 × 2 identity matrix. Thus we have
+�����
+−β
+1
+µ
+−3ℓ2
+µr4 + 2K
+r3 (1 + α)
+−β
+����� = 0
+(23)
+The characteristic equation (the eigenvalue equation) becomes:
+β1,2 = 1
+2
+�
+τ ±
+�
+τ 2 − 4∆
+�
+(24)
+τ = trace(J) = 0
+(25)
+∆ = det(J) = µ2K4(1 + α)4
+ℓ6
+(26)
+Following [29], the stability is determined by the sign of the eigenvalues. Since ∆ > 0, we have
+the following cases:
+• τ < 0, τ 2 − 4∆ > 0 ⇒ (r0, pr0) a stable node.
+• τ < 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) a stable spiral.
+6
+
+• τ > 0, τ 2 − 4∆ > 0 ⇒ (r0, pr0) an unstable node.
+• τ > 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) an unstable spiral.
+• τ = 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) a neutrally stable center (which is our case).
+Actually, the stability refers to how the solution behaves near the equilibrium point; in that
+unstable solutions grow to infinity, whereas stable solutions tend to zero. Also, it is the imaginary
+cases which are the ones giving bound orbital solutions (specifically the center case, whereas the
+stable and unstable imaginary cases are bound solutions tending towards or away from zero).
+3
+Stability & Bertrands theorem
+First, we rewrite the eigenvalue equation in the form
+β2 + µ2K4(1 + α)4
+ℓ6
+= 0
+(27)
+leading to:
+β = ±iµK2(1 + α)2
+ℓ3
+(28)
+We note that one can study the case for a purely Newtonian Potential by letting α → 0.
+Similarly, by ignoring the terms derived from the Newtonian potential, one can single out the
+pure Yukawa contribution. In these two extreme cases, the characteristic equations becomes
+Pure Newtonian: β2 + µ2K4
+ℓ6
+= 0
+(29)
+Pure Yukawa: β2 + µ2K4α4
+ℓ6
+= 0
+(30)
+giving
+Pure Newtonian: β = ±iµK2
+ℓ3
+(31)
+Pure Yukawa: β = ±iµK2α2
+ℓ3
+(32)
+Thus, the equilibrium points for the purely Newtonian, the purely Yukawa, and the Newton
+plus Yukawa Potentials remain center solutions. This implies that the motion would remain
+restricted to ellipses about the equilibrium point; and so, orbits near the equilibrium point are
+possible (further away from the equilibrium point one would have unbounded solutions, as Fig.
+1 shows). This proves that for small r/λ we have stable, closed orbits.
+For the Keplerian orbit equation, it can be written as:
+d2u
+dϕ2 + u = − µ
+ℓ2
+d
+duV
+�1
+u
+�
+(33)
+where u = 1r denotes the Binet transformation, giving, for small r/λ, the following differential
+equation:
+d2u
+dϕ2 + u = +µK
+ℓ2 (1 + α)
+(34)
+whose solution is given by
+u(ϕ) = 1
+r = A [1 + e cos (ϕ − ϕ0)] : A = µK
+ℓ2 (1 + α)
+(35)
+7
+
+with e is the eccentricity of the orbit. The purely Newtonian and purely Yukawa cases follow
+respectively from (34)
+Newtonoian: u(ϕ) = 1
+r = µK
+ℓ2 [1 + e cos (ϕ − ϕ0)]
+(36)
+Purely Yukawa: u(ϕ) = 1
+r = µKα
+ℓ2
+[1 + e cos (ϕ − ϕ0)]
+(37)
+Finally, in order to satisfy Bertrands theorem, the following condition should be satisfied
+d2Vred(r)
+dr2
+����
+r=r0
+> 0
+(38)
+where the reduced potential is given by (16). With the approximations of (eq. 18)) and ignoring
+terms of order O
+�
+r2
+λ2
+�
+this condition becomes
+d2Vred(r)
+dr2
+����
+r=r0
+= µ2K4(1 + α)4
+ℓ6
+> 0
+(39)
+which is true, since α, µ, K, ℓ > 0, in general and in the special cases of Newtonian (α = 0)
+and purely Yukawa potentials. This shows that the Yukawa plus Newtonian potential satisfies
+Bertrands theorem for small rλ.
+4
+Application to the solar system
+We present here our results consisting of determining first the parameters of the models (rmin, rmax, a, b, e)
+by comparing the previous approximative analytical solutions with the NASA data. Then, we
+solved the equations of motion numerically using Matlab and the fourth-order Runge-Kutta
+method with no approximation so that to be compared with the analytical solutions and with
+the observed NASA data. We applied this for all the planets of the solar system. For each pair
+(sun-planet) we used the following values M⊙ = 1.9885 × 1030kg, αnominal = 10−8, λ = 1015m.
+We list in Table (1) the initial conditions used in the analytical and numerical calculations (the
+period τ is used only in the numerical solution to determine the corresponding ‘step’):
+MERCURY
+VENUS
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+NEPTUNE
+PLUTO
+mp(×1024kg)0.3302
+4.8673
+5.9722
+0.64169
+1898.13
+568.32
+86.811
+102.409
+0.01303
+τ (days)
+87.969
+224.701
+365.256
+686.98
+4332.589
+10832.33
+30685.4
+60189
+90560
+rmin
+(×106km)
+0.046
+0.10748
+0.147095
+0.20665
+0.740595
+1.357554
+2.732696
+4.47105
+4.434987
+vmax
+(×103m/s)
+58.98
+35.26
+30.29
+26.5
+13.72
+10.18
+7.11
+5.5
+6.1
+eccentricity
+0.20563
+0.00677
+0.01671
+0.09341
+0.04839
+0.05415
+0.04717
+0.00859
+0.24881
+Table 1: Initial conditions used in the calculations where mp denotes the planet mass, τ is the
+orbit period, rmin is the perihelion and vmax denotes the perihelion velocity
+4.1
+Analytical Method
+The analytical ellipsis equation is of the form
+1
+r ≡ u
+=
+a
+b2 (1 + e cos ϕ) ,
+(40)
+8
+
+where (for a y-axis perpendicular to the polar axis in the orbit plane)
+rmin = a(1 − e)
+,
+rmax = a(1 + e),
+(41)
+e = c
+a =
+�
+1 − b2
+a2
+:
+c2 = a2 − b2,
+(42)
+a = rmin + rmax
+2
+,
+b = ymax − ymin
+2
+(43)
+Thus, analytically one can start with (rmin, vmax, e) observed by NASA in [24] to compute†:
+a = rmin
+1 − e
+,
+b = a
+�
+1 − e2,
+(44)
+and estimate the strength α from
+µK
+ℓ2 (1 + α) = a
+b2
+using
+ℓ = rminvmax.
+(45)
+Once the analytical equation is determined, then one can plot the trajectory and recompute the
+characteristics (rmin, rmax, a, b, e) using Eqs (41,42). We call this procedure the “analytical-α-
+estimated” approach.
+One can also use the nominal value of α = 10−8, and plug it in Eq. (40), where ℓ, e are taken
+from the observed data, to re-evaluate (rmin, rmax, a, b, e) from
+a =
+1
+A(1 − e2),
+A = µK(1+α)
+ℓ2
+,
+b =
+1
+A
+√
+1 − e2 .
+(46)
+We call this procedure the “analytical-α-nominal” approach, which can be looked at as a method
+with three inputs (α, ℓ, e) instead of the three inputs (rmin, vmax, e) used in the other approach.
+4.2
+Numerical Method
+Here, we just solve numerically, using the fourth-order Runge-Kutta method, the Newton’s law
+equation of motion in the c.o.m frame with initial conditions taken from NASA. Thus we solve
+the equations:
+¨⃗r1 = Gmp
+⃗r2 − ⃗r1
+r3
+, Newton,
+¨⃗r2 = GM⊙
+⃗r1 − ⃗r2
+r3
+= −M⊙
+mp
+¨⃗r1,
+(47)
+¨⃗r1 = Gmp
+�
+(1 + αe− r
+λ )1
+r + α
+λe− r
+λ
+� ⃗r2 − ⃗r1
+r2
+, Newton+Yukawa,
+¨⃗r2 = −M⊙
+mp
+¨⃗r1,
+(48)
+under the initial conditions given by NASA data of (rmin, vmax):
+⃗r1(t = tmin) =
+mp
+mp + M⊙
+⃗rmin
+,
+⃗v1(t = tmin) =
+mp
+mp + M⊙
+⃗vmax,
+(49)
+⃗r2(t = tmin) = −
+M⊙
+mp + M⊙
+⃗rmin
+,
+⃗v2(t = tmin) = −
+M⊙
+mp + M⊙
+⃗vmax.
+(50)
+Once the trajectory is solved numerically, we check that it is closed, as the Fig.
+(2) shows
+for both the pure Newton and that with the Yukawa corrections (since the differences are not
+visible on the figure scale containing all the planets). For each obtained orbit, we recalculate
+the corresponding characteristics (rmin, rmax, a, b, e).
+†Due to measurement errors and orbits not being perfectly elliptical, the NASA data may give slightly different
+values of a using Eq. 43 or Eq. 44.
+9
+
+Figure 2: Closed bound planets’ trajectories with and without Yukawa corrections with strength
+α nominal.
+4.3
+Results
+We report in the Tables of Appendix A (from A1 to A18), the calculated characteristics of the
+resulting trajectories for all the planets in the solar system, corresponding to the pure Newton
+and the Newton corrected with Yukawa potentials, both in the analytical and the numerical
+approaches. The odd (even) numbered tables correspond to the nominal (estimated) Yukawa
+strength α. The number of moons of each planet is determined according to [30]. Below we
+explain the meanings of the symbols used in the tables.
+• Nnum: Numerical calculations using the Newtonian potential.
+• Nanal: Analytical calculations using the Newtonian potential.
+• RN = Nnum
+Nanal %: The percentage ratio of the numerical to the analytical results for Newton
+potential.
+• (N + Y K)num : Numerical calculations using the modified potential.
+• (N + Y K)anal: Analytical calculations using the modified potential.
+• RN+Y K = (N+Y K)num
+(N+Y K)anal %: The percentage ratio of the numerical to the analytical results
+for modified potential.
+• RN−Obs
+num
+= Nnum/Obs%: Percentage ratio of the numerical results, using the Newtonian
+potential, to the observed results.
+• RN−Obs
+anal
+= Nanal/Obs %: Percentage ratio the analytical results, using the Newtonian
+potential, to the observed results.
+• RY K−Obs
+num
+= (N + Y K)num/Obs %: Percentage ratio of the numerical results, using the
+modified potential, to the observed results.
+10
+
+(km)• RY K−Obs
+anal
+= (N + Y K)anal/Obs %: Percentage ratio of the analytical results, using the
+modified potential, to the observed results.
+In order to summarize the findings of the Tables, we present in Fig. (3) plots showing, for each
+planet and at every polar angle, the deviation from unity of the ratio between two quantities of
+the following, allowing thus to compare the effects of the considered potential (Newton vs New-
+ton+Yuakawa) and/or the used method (numerical vs analytical) and/or the Yukawa strength
+determination (nominal vs estimated):
+• rn(num) representing the trajectory equation of the numerical approach with Newton
+potential,
+• rn(anl) representing the trajectory equation of the analytical approach with Newton po-
+tential,
+• ryk(num) representing the trajectory equation of the numerical approach with New-
+ton+Yukawa potential and nominal α,
+• ryk(anl) representing the trajectory equation of the analytical approach with Newton+Yukawa
+potential and nominal α,
+• ryka(num) representing the trajectory equation of the numerical approach with New-
+ton+Yukawa potential and estimated α,
+• ryka(anl) representing the trajectory equation of the analytical approach with New-
+ton+Yukawa potential and estimated α.
+We see that some ratios (e.g. the dashed red and sky blue) do coincide near zero deviation
+from one, meaning no tangible effect of adding the Yukawa correction, be it in the analytic or
+the numeric method, as long as one takes the nominal value of α. Also,we note local extremums
+for the deviations from unity at polar angles multiples of π/2 as a generic feature in many plots.
+One can interpret the large values of the deviations for the nearest (Mercury) and the farthest
+(Pluto) planet, in that for the former; the perturbative effect of solar winds, important as we
+approach the sun, was not taken into consideration, whereas for the furthest; accumulating
+gravitational screening effects of the other planets and their moons, which were not considered
+in the study, are becoming important especially for a small sized- planetoid like Pluto.
+In order to show the effects of the separating distance effect, one should compute the absolute
+deviations from observed data for each planet. In appendix B, the Tables B1, B2 (B3, B4), report
+the deviation from observation for each planet of rmax, rmin respectively, in the case of nominal
+(estimated) α. We summarize these findings in Fig. (4). We see that the agreement between
+the numerical and analytical solutions is excellent in both estimated and nominal α cases. We
+see that the deviations due to Yukawa correction are not large, but note the following:
+1. For estimated α:
+• rmax-deviation: The numerical deviation is larger by about 103 times the analytical
+deviation. In general, it increases the further the planet is, and reaches a maximum
+of order (−25 million km) (less than the observed value) in Pluto.
+• rmin-deviation: Again, the numerical deviation is larger by about (101-102)-order of
+magnitude than analytical deviation, where it is largest in Neptune (−8 million km),
+however it reverses sign and becomes (+5 million km) more than the observation in
+Pluto.
+11
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  1-{ rn(num)/ryka(anl) }        ____ 
+1-{ryka(anl)/ryk(num)}      ____ 
+1-{ryka(anl)/ryka(num)}    ____ 
+ 1-{ rn(anl)/ryka(anl)}           ____ 
+1-{ryka(anl)/ryk(anl)}         ____ 
+1-{ rn(num)/ryk(num)}    ----- 
+1-{ryka(num)/rn(num)}   ----- 
+1-{ryk(num)/ryka(num)} ----- 
+1-{ryk(anl)/rn(anl)}          ----- 
+ 
+ 
+Figure 3:
+Deviations from Unity for Ratios of Computed trajectories at each polar angle,
+according to the considered potential (Newton vs Newton+Yuakawa) and/or to the used method
+(numerical vs analytical) and/or to the Yukawa strength determination (nominal vs estimated).
+We show in a zoomed region, for one planet (Earth) generic case, that the dashed red and sky
+blue curves are very near each other (the same applies to the green and blue curves in Mercury
+case).
+12
+
+VENUS
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+0.03
+0.02
+1-Percentage Ratio(%)
+0.01
+0.01
+-0.02
+-0.03
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)EARTH
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+0.08
+0.06
+1-Percentage Ratio(%)
+0.04
+0.02
+0.02
+-0.04
+-0.06
+-0.08
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)MARS
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+2
+1.5
+0.5
+0.5
+1
+-1.5
+-2
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)JUPITER
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+0.6
+0.4
+1-Percentage Ratio(%)
+0.2
+-0.2
+-0.4
+-0.6
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)SATURN
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+3
+2
+-1
+-2
+-3
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)URANUS
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+1.5
+1-Percentage Ratio(%)
+0.5
+0.5
+1
+-1.5
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)Neptune
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+2 
+1.5
+0.5
+0.5
+-1.5
+-2
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)Pluto
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+15
+10
+1-Percentage Ratio(%)
+-5
+-10
+-15
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)MERCURY
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+10
+8
+6
+1-PercentageRatio(%)
+4
+N
+A
+-6
+-8
+-10
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg) 
+ 
+ 
+ 
+ 
+-50
+0
+50
+100
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmax; alpha nominal
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+-20
+0
+20
+40
+60
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmin; alpha nominal
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+-30
+-20
+-10
+0
+10
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmax; alpha estimated
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+-10
+-5
+0
+5
+10
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmin; alpha estimated
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+Figure 4: Absolute deviations from observed data for each planet, according to the used method
+(numerical vs analytical) and/or to the Yukawa strength determination (nominal vs estimated).
+13
+
+2. For nominal α:
+• rmax-deviation: The numerical deviation is larger than the analytical one, but are of
+the same order reaching a maximum of +80 (+40) million km using the numerical
+(analytical) method in Neptune. For Pluto and Uranus, we get (−40) million km in
+the numerical method (less than observed).
+• rmin-deviation: The analytical deviation is larger, and sometimes reverses sign com-
+pared to the numerical. For example, in Neptune the analytical approach gives a
+deviation of (+40 million km) from observation, whereas the numerical one gives a
+deviation of −5 million km (less than the observed value).
+Actually, the disagreements with observations are due to several reasons: the first one is physical
+in nature, in that it results from neglecting the perturbation due to third bodies, or, more
+generally, the effect of the natural satellites, such as moons or asteroids.
+Also, we did not
+either take into account the radiation and the solar wind physical effects. Moreover, the results
+were obtained as a 2-body problem, and hence the movement of more distant planets might be
+affected by planets closer to the sun, which can be present not in the dominant term, but in
+higher orders of expansion. The second factor lies in the computational side, and concerns the
+numerical method used, the value of the step size, and the high sensitivity of the problem to
+the initial conditions. One should also mention that for the analytical solution we restricted the
+study to leading order neglecting higher orders in the expansion of exponentials, whereas for the
+numerical solution the entire exponential is considered.
+5
+Summary and Conclusion
+In this work, we followed [23] and used the Hamilton’s formulation in order to obtain the
+differential equation of motion and the path equation for the gravitational two-body system.
+The developments are carried out in the case of the pure Newtonian potential, the Newtonian
+corrected with Yukawa type potential and the pure Yukawa potential.
+As in [23], we have
+reviewed the stability problem, constructed the linearization matrix and tested the stability
+of the system for a Yukawa correction, and found that it is of a central solution type, which
+implies stable solutions near the fixed point. We repeated the analysis for a purely Yukawa force
+and found similar results. We also confirmed that the modified potential obeys the Bertrands
+theorem.
+Then, we determined the parameters’ set corresponding to the planets of the solar system
+starting from the observed (rmin, vmax, e) estimating α. For both the estimated and nominal
+values of α, we determined the characteristics of the trajectories numerically and analytically,
+and compared between the methods and with the observed data. We explained the extent to
+which these results are consistent with the observational data, presenting in form of histograms
+the absolute deviations from observations, which were found to give an upper deviation of order
+80 million km in Neptune using nominal α, and 20 million km in Pluto using estimated α.
+Acknowledgments:
+N. Chamoun acknowledges support from the ICTP-Associate pro-
+gram, from the Humboldt Foundation and from the CAS-PIFI scholarship.
+14
+
+Appendices
+A. Tables of Calculated/Observed Parameters of the Planets
+15
+
+Mercury
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+46
+69.832
+57.916
+56.67679158
+0.205744228
+Nanal
+47
+72.043
+59.756
+58.47840586
+0.205646344
+RN = Nnum
+Nanal %
+97
+96.930
+96.921
+96.91918025
+99.95242442
+(N + Y K)num
+46
+69.831
+57.916
+56.67678243
+0.205738221
+(N + Y K)anal
+47
+72.043
+59.756
+58.47840528
+0.205646344
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+97
+96.930
+96.921
+96.91916556
+99.95534277
+Observation
+46
+69.818
+57.909
+0.20563069
+RN−Obs
+num
+= Nnum/Obs %
+100
+99.980
+99.988
+99.94481595
+RN−Obs
+anal
+= Nanal/Obs %
+97
+96.911
+96.909
+99.9923879
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100
+99.981
+99.988
+99.94773407
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+97
+96.911
+96.909
+99.9923879
+nominal α = 10−8
+Table A1: The values of the calculated and observational astronomical parameters of the planet
+Mercury whose number of moons is 0
+Mercury
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+46
+69.623
+57.826
+56.65144795
+0.2022
+Nanal
+46
+69.819
+57.912
+56.67470066
+0.2056
+RN = Nnum
+Nanal %
+100
+99.719
+99.851
+99.95897162
+98.3743
+(N + Y K)num
+46
+69.613
+57.820
+56.64729064
+0.2022
+(N + Y K)anal
+46
+69.815
+57.908
+56.6714469
+0.2056
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100
+99.711
+99.847
+99.9573749
+98.3408
+Observation
+46
+69.818
+57.909
+0.2056
+RN−Obs
+num
+= Nnum/Obs %
+100
+99.721
+99.856
+98.3817
+RN−Obs
+anal
+= Nanal/Obs %
+100
+100.001
+100.005
+100.0075
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100
+99.706
+99.847
+98.3482
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100
+99.995
+99.999
+100.0075
+estimated α = 5.741444131301954 × 10−5
+Table A2: The values of the calculated and observational astronomical parameters of the planet
+Mercury whose number of moons is 0
+16
+
+Venus
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+107.30
+108.689
+107.99
+107.9982364
+0.0044
+Nanal
+107.48
+108.961
+108.22
+108.2222348
+0.0072
+RN = Nnum
+Nanal %
+99.83
+99.750
+99.79
+99.79301998
+60.8187
+(N + Y K)num
+107.30
+108.689
+107.99
+107.9982353
+0.0044
+(N + Y K)anal
+107.48
+108.961
+108.22
+108.2222337
+0.0072
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.83
+99.750
+99.79
+99.79301998
+60.8189
+Observation
+107.48
+108.941
+108.21
+0.0068
+RN−Obs
+num
+= Nnum/Obs %
+99.83
+99.769
+99.80
+64.7150
+RN−Obs
+anal
+= Nanal/Obs %
+100.00
+100.018
+100.01
+106.4063
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.83
+99.769
+99.80
+64.7152
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.00
+100.018
+100.01
+106.4063
+nominal α = 10−8
+Table A3: The values of the calculated and observational astronomical parameters of the planet
+Venus whose number of moons is 0
+Venus
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+107.30
+108.689
+107.99
+107.9982364
+0.0044
+Nanal
+107.48
+108.961
+108.22
+108.2222348
+0.0072
+RN = Nnum
+Nanal %
+99.83
+99.750
+99.79
+99.79301998
+60.8187
+(N + Y K)num
+107.30
+108.658
+107.98
+107.9821956
+0.0045
+(N + Y K)anal
+107.47
+108.945
+108.20
+108.2068155
+0.0072
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.84
+99.736
+99.78
+99.79241613
+63.0300
+Observation
+107.48
+108.941
+108.21
+0.0068
+RN−Obs
+num
+= Nnum/Obs %
+99.83
+99.769
+99.80
+64.7150
+RN−Obs
+anal
+= Nanal/Obs %
+100.00
+100.018
+100.01
+106.4063
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.83
+99.740
+99.78
+67.0680
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.99
+100.004
+99.99
+106.4063
+estimated α = 1.424988220126711 × 10−4
+Table A4: The values of the calculated and observational astronomical parameters of the planet
+Venus whose number of moons is 0
+17
+
+EARTH
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+146.884
+151.7
+149.336
+149.319847
+0.0156
+Nanal
+147.126
+152.1
+149.625
+149.6034965
+0.0168
+RN = Nnum
+Nanal %
+99.835
+99.7
+99.806
+99.81039915
+92.5721
+(N + Y K)num
+146.884
+151.7
+149.336
+149.3198455
+0.0156
+(N + Y K)anal
+147.126
+152.1
+149.625
+149.603495
+0.0168
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.835
+99.7
+99.806
+99.81039915
+92.5720
+Observation
+147.095
+152.1
+149.598
+0.0167
+RN−Obs
+num
+= Nnum/Obs %
+99.8572
+99.7
+99.825
+93.5903
+RN−Obs
+anal
+= Nanal/Obs %
+99.978
+99.9
+99.981
+98.9120
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.857
+99.7
+99.825
+93.5902
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.978
+99.9
+99.981
+98.9120
+nominal α = 10−8
+Table A5: The values of the calculated and observational astronomical parameters of the planet
+Earth whose number of moons is 0
+EARTH
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+146.884
+151.7
+149.336
+149.319847
+0.0156
+Nanal
+147.126
+152.1
+149.625
+149.6034965
+0.0168
+RN = Nnum
+Nanal %
+99.835
+99.7
+99.806
+99.81039915
+92.5721
+(N + Y K)num
+146.883
+151.7
+149.307
+149.2910008
+0.0154
+(N + Y K)anal
+147.099
+152.0
+149.597
+149.5762082
+0.01688
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.853
+99.7
+99.805
+99.80932302
+91.4700
+Observation
+147.095
+152.1
+149.598
+0.0167
+RN−Obs
+num
+= Nnum/Obs %
+99.8572
+99.7
+99.825
+93.5903
+RN−Obs
+anal
+= Nanal/Obs %
+99.978
+99.9
+99.981
+98.9120
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.856
+99.7
+99.805
+92.4761
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.003
+99.9
+99.999
+101.0999
+estimated α = 1.824376359731428 × 10−4
+Table A6: The values of the calculated and observational astronomical parameters of the planet
+Earth whose number of moons is 0
+18
+
+MARS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+206.57
+248.480
+227.52
+226.6509159
+0.0898
+Nanal
+206.64
+249.277
+227.96
+226.9631182
+0.0935
+RN = Nnum
+Nanal %
+99.96
+99.680
+99.80
+99.8624436
+96.0965
+(N + Y K)num
+206.57
+248.480
+227.52
+226.6509134
+0.0898
+(N + Y K)anal
+206.64
+249.277
+227.96
+226.9631159
+0.0935
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.96
+99.680
+99.80
+99.86244351
+96.0965
+Observation
+206.65
+249.261
+227.94
+0.0935
+RN−Obs
+num
+= Nnum/Obs %
+99.96
+99.687
+99.81
+96.1252
+RN−Obs
+anal
+= Nanal/Obs %
+99.99
+100.006
+100.01
+100.0298
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.96
+99.687
+99.81
+96.1252
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.99
+100.006
+100.01
+100.0298
+nominal α = 10−8
+Table A7: The values of the calculated and observational astronomical parameters of the planet
+Mars whose number of moons is 0
+MARS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+206.57
+248.480
+227.52
+226.6509159
+0.0898
+Nanal
+206.64
+249.277
+227.96
+226.9631182
+0.0935
+RN = Nnum
+Nanal %
+99.96
+99.680
+99.80
+99.8624436
+96.0965
+(N + Y K)num
+206.57
+248.425
+227.49
+226.6249874
+0.0897
+(N + Y K)anal
+206.62
+249.252
+227.93
+226.9402451
+0.0935
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.97
+99.668
+99.80
+99.86108339
+95.9861
+Observation
+206.65
+249.261
+227.94
+0.0935
+RN−Obs
+num
+= Nnum/Obs %
+99.96
+99.687
+99.81
+96.1252
+RN−Obs
+anal
+= Nanal/Obs %
+99.99
+100.006
+100.01
+100.0298
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.96
+99.664
+99.80
+96.0147
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.98
+99.996
+99.99
+100.0298
+estimated α = 1.007889331583467 × 10−4
+Table A8: The values of the calculated and observational astronomical parameters of the planet
+Mars whose number of moons is 0
+19
+
+JUPITER
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+739.902
+815.533
+777.717
+776.9190412
+0.0469
+Nanal
+742.542
+818.568
+780.555
+779.626266
+0.04873
+RN = Nnum
+Nanal %
+99.644
+99.629
+99.636
+99.65275352
+96.3711
+(N + Y K)num
+739.902
+815.533
+777.717
+776.9190329
+0.0469
+(N + Y K)anal
+742.542
+818.568
+780.555
+779.6262582
+0.0487
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.644
+99.629
+99.636
+99.65275345
+96.3711
+Observation
+740.595
+816.363
+778.479
+0.0487
+RN−Obs
+num
+= Nnum/Obs %
+99.906
+99.898
+99.902
+96.4399
+RN−Obs
+anal
+= Nanal/Obs %
+100.262
+100.270
+100.266
+100.0714
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.906
+99.898
+99.902
+96.4399
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.262
+100.270
+100.266
+100.0714
+nominal α = 10−8
+Table A9: The values of the calculated and observational astronomical parameters of the planet
+Jupiter whose number of moons is 0
+JUPITER
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+739.902
+815.533
+777.717
+776.9190412
+0.0469
+Nanal
+742.542
+818.568
+780.555
+779.626266
+0.04873
+RN = Nnum
+Nanal %
+99.644
+99.629
+99.636
+99.65275352
+96.3711
+(N + Y K)num
+739.837
+810.932
+775.385
+774.6852056
+0.0441
+(N + Y K)anal
+740.567
+816.390
+778.478
+777.5526264
+0.0487
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.901
+99.331
+99.602
+99.63122486
+90.6263
+Observation
+740.595
+816.363
+778.479
+0.0487
+RN−Obs
+num
+= Nnum/Obs %
+99.906
+99.898
+99.902
+96.4399
+RN−Obs
+anal
+= Nanal/Obs %
+100.262
+100.270
+100.266
+100.0714
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.897
+99.334
+99.602
+90.6911
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.996
+100.003
+99.999
+100.0714
+estimated α = 2.666880127522 × 10−3
+Table A10: The values of the calculated and observational astronomical parameters of the planet
+Jupiter whose number of moons is 0
+20
+
+SATURN
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+1355.461
+1523.344
+1439.403
+1437.455093
+0.055
+Nanal
+1368.378
+1518.496
+1443.437
+1441.481829
+0.052
+RN = Nnum
+Nanal %
+99.056
+100.319
+99.720
+99.72065302
+106.042
+(N + Y K)num
+1355.461
+1523.344
+1439.403
+1437.455078
+0.055
+(N + Y K)anal
+1368.378
+1518.496
+1443.437
+1441.481815
+0.052
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.056
+100.319
+99.720
+99.72065294
+106.042
+Observation
+1357.554
+1506.527
+1432.041
+0.052
+RN−Obs
+num
+= Nnum/Obs %
+99.845
+101.116
+100.514
+106.081
+RN−Obs
+anal
+= Nanal/Obs %
+100.797
+100.794
+100.795
+100.036
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.845
+101.116
+100.514
+106.081
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.797
+100.794
+100.795
+100.036
+nominal α = 10−8
+Table A11: The values of the calculated and observational astronomical parameters of the planet
+Saturn whose number of moons is 0
+SATURN
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+1355.461
+1523.344
+1439.403
+1437.455093
+0.055
+Nanal
+1368.378
+1518.496
+1443.437
+1441.481829
+0.052
+RN = Nnum
+Nanal %
+99.056
+100.319
+99.720
+99.72065302
+106.042
+(N + Y K)num
+1354.869
+1497.652
+1426.261
+1424.954776
+0.046
+(N + Y K)anal
+1357.574
+1506.507
+1432.040
+1430.100672
+0.052
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.800
+99.412
+99.596
+99.64017246
+89.244
+Observation
+1357.554
+1506.527
+1432.041
+0.052
+RN−Obs
+num
+= Nnum/Obs %
+99.845
+101.116
+100.514
+106.081
+RN−Obs
+anal
+= Nanal/Obs %
+100.797
+100.794
+100.795
+100.036
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.802
+99.410
+99.596
+89.277
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.001
+99.998
+99.999
+100.036
+estimated α = 7.958291053541 × 10−3
+Table A12: The values of the calculated and observational astronomical parameters of the planet
+Saturn whose number of moons is 0
+21
+
+URANUS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+2729.595
+2957.44
+2843.519
+2841.649275
+0.0381
+Nanal
+2717.213
+2984.63
+2850.921
+2847.766462
+0.0469
+RN = Nnum
+Nanal %
+100.455
+99.08
+99.740
+99.78519352
+81.2504
+(N + Y K)num
+2729.595
+2957.44
+2843.519
+2841.649245
+0.0381
+(N + Y K)anal
+2717.213
+2984.63
+2850.921
+2847.766434
+0.0469
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100.455
+99.08
+99.740
+99.78519344
+81.2504
+Observation
+2732.696
+3001.39
+2867.043
+0.0469
+RN−Obs
+num
+= Nnum/Obs %
+99.886
+98.53
+99.179
+81.3684
+RN−Obs
+anal
+= Nanal/Obs %
+99.433
+99.44
+99.437
+100.1452
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.886
+98.53
+99.179
+81.3684
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.433
+99.44
+99.437
+100.1452
+nominal α = 10−8
+Table A13: The values of the calculated and observational astronomical parameters of the planet
+Uranus whose number of moons is 0
+URANUS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+2729.595
+2957.44
+2843.519
+2841.649275
+0.0381
+Nanal
+2717.213
+2984.63
+2850.921
+2847.766462
+0.0469
+RN = Nnum
+Nanal %
+100.455
+99.08
+99.740
+99.78519352
+81.2504
+(N + Y K)num
+2730.116
+2992.91
+2861.516
+2858.935401
+0.0441
+(N + Y K)anal
+2732.578
+3001.50
+2867.042
+2863.869614
+0.0469
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.909
+99.71
+99.807
+99.82770818
+94.0932
+Observation
+2732.696
+3001.39
+2867.043
+0.0469
+RN−Obs
+num
+= Nnum/Obs %
+99.886
+98.53
+99.179
+81.3684
+RN−Obs
+anal
+= Nanal/Obs %
+99.433
+99.44
+99.437
+100.1452
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.905
+99.71
+99.807
+94.2299
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.995
+100.00
+99.999
+100.1452
+estimated α = −5.622864957252 × 10−3
+Table A14: The values of the calculated and observational astronomical parameters of the planet
+Uranus whose number of moons is 0
+22
+
+Neptune
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4464.81
+4634.099
+4549.454
+4548.810665
+0.0177
+Nanal
+4512.97
+4601.381
+4557.176
+4556.953752
+0.0097
+RN = Nnum
+Nanal %
+98.93
+100.711
+99.830
+99.82130416
+180.8744
+(N + Y K)num
+4464.81
+4634.098
+4549.454
+4548.810617
+0.0177
+(N + Y K)anal
+4512.97
+4601.381
+4557.176
+4556.953706
+0.0097
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+98.93
+100.711
+99.830
+99.82130411
+180.8743
+Observation
+4471.05
+4558.857
+4514.953
+0.0097
+RN−Obs
+num
+= Nnum/Obs %
+99.86
+101.650
+100.764
+182.6474
+RN−Obs
+anal
+= Nanal/Obs %
+100.93
+100.932
+100.935
+100.9802
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.86
+101.650
+100.764
+182.6473
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.93
+100.932
+100.935
+100.9802
+nominal α = 10−8
+Table A15: The values of the calculated and observational astronomical parameters of the planet
+Neptune whose number of moons is 0
+Neptune
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4464.81
+4634.099
+4549.454
+4548.810665
+0.0177
+Nanal
+4512.97
+4601.381
+4557.176
+4556.953752
+0.0097
+RN = Nnum
+Nanal %
+98.93
+100.711
+99.830
+99.82130416
+180.8744
+(N + Y K)num
+4463.01
+4546.479
+4504.745
+4504.517794
+0.0096
+(N + Y K)anal
+4471.15
+4558.747
+4514.952
+4514.73215
+0.0097
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.81
+99.730
+99.773
+99.77375499
+98.6587
+Observation
+4471.05
+4558.857
+4514.953
+0.0097
+RN−Obs
+num
+= Nnum/Obs %
+99.86
+101.650
+100.764
+182.6474
+RN−Obs
+anal
+= Nanal/Obs %
+100.93
+100.932
+100.935
+100.9802
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.82
+99.728
+99.773
+99.6259
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.00
+99.997
+99.999
+100.9802
+estimated α = 9.351961741362 × 10−3
+Table A16: The values of the calculated and observational astronomical parameters of the planet
+Neptune whose number of moons is 0
+23
+
+Pluto
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4439.709
+7265.423
+5852.566
+5684.326067
+0.2397
+Nanal
+4431.722
+7298.614
+5865.168
+5687.267307
+0.2444
+RN = Nnum
+Nanal %
+100.180
+99.545
+99.785
+99.94828377
+98.0832
+(N + Y K)num
+4439.709
+7265.423
+5852.566
+5684.325992
+0.2397
+(N + Y K)anal
+4431.722
+7298.614
+5865.168
+5687.26725
+0.2444
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100.180
+99.545
+99.785
+99.94828346
+98.0832
+Observation
+4434.987
+7304.326
+5869.656
+0.2444
+RN−Obs
+num
+= Nnum/Obs %
+100.106
+99.467
+99.708
+98.0882
+RN−Obs
+anal
+= Nanal/Obs %
+99.926
+99.921
+99.923
+100.0051
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100.106
+99.467
+99.708
+98.0882
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.926
+99.921
+99.923
+100.0051
+nominal α = 10−8
+Table A17: The values of the calculated and observational astronomical parameters of the planet
+Pluto whose number of moons is 0
+Pluto
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4439.709
+7265.423
+5852.566
+5684.326067
+0.2397
+Nanal
+4431.722
+7298.614
+5865.168
+5687.267307
+0.2444
+RN = Nnum
+Nanal %
+100.180
+99.545
+99.785
+99.94828377
+98.0832
+(N + Y K)num
+4439.740
+7280.242
+5859.991
+5690.112819
+0.2407
+(N + Y K)anal
+4435.112
+7304.196
+5869.654
+5691.616958
+0.2444
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100.104
+99.672
+99.835
+99.97357273
+98.4812
+Observation
+4434.987
+7304.326
+5869.656
+0.2444
+RN−Obs
+num
+= Nnum/Obs %
+100.106
+99.467
+99.708
+98.0882
+RN−Obs
+anal
+= Nanal/Obs %
+99.926
+99.921
+99.923
+100.0051
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100.107
+99.670
+99.835
+98.4862
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.002
+99.998
+99.999
+100.0051
+estimated α = −7.642205983339201 × 10−4
+Table A18: The values of the calculated and observational astronomical parameters of the planet
+Pluto whose number of moons is 0
+24
+
+B. Tables of Absolute Deviations from Observation of the Planets
+25
+
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmax
+rnum
+max − Obs
+ranal
+max − Obs
+MERCURY
+99.721
+100.001
+69.818
+-0.194
+0.001
+Venus
+99.769
+100.018
+108.941
+-0.251
+0.020
+EARTH
+99.794
+99.984
+152.100
+-0.312
+-0.024
+MARS
+99.687
+100.006
+249.261
+-0.780
+0.016
+JUPITER
+99.898
+100.270
+816.363
+-0.829
+2.205
+SATURN
+101.116
+100.794
+1506.527
+16.817
+11.969
+URANUS
+98.535
+99.441
+3001.390
+-43.947
+-16.759
+Neptune
+101.650
+100.932
+4558.857
+75.241
+42.524
+Pluto
+99.467
+99.921
+7304.326
+-38.902
+-5.711
+Table B1: Absolute deviations, with nominal α, of rmax from observation, evaluated in (106
+km).
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmin
+rnum
+min − Obs
+ranal
+min − Obs
+MERCURY
+100.062
+100.012
+46.000
+0.028
+0.005
+Venus
+99.839
+100.008
+107.480
+-0.172
+0.009
+EARTH
+99.857
+99.978
+147.095
+-0.210
+-0.031
+MARS
+99.962
+99.999
+206.650
+-0.077
+-0.001
+JUPITER
+99.906
+100.262
+740.595
+-0.692
+1.947
+SATURN
+99.845
+100.797
+1357.554
+-2.092
+10.824
+URANUS
+99.886
+99.433
+2732.696
+-3.100
+-15.482
+Neptune
+99.860
+100.937
+4471.050
+-6.239
+41.922
+Pluto
+100.106
+99.926
+4434.987
+4.722
+-3.264
+Table B2: Absolute deviations, with nominal α, of rmin from observation, evaluated in (106 km).
+26
+
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmax
+rnum
+max − Obs
+ranal
+max − Obs
+MERCURY
+99.706
+99.995
+69.818
+-0.204
+-0.002
+Venus
+99.740
+100.004
+108.941
+-0.282
+0.004
+EARTH
+99.757
+99.997
+152.100
+-0.369
+-0.003
+MARS
+99.664
+99.996
+249.261
+-0.835
+-0.008
+JUPITER
+99.334
+100.003
+816.363
+-5.430
+0.027
+SATURN
+99.410
+99.998
+1506.527
+-8.874
+-0.019
+URANUS
+99.717
+100.003
+3001.390
+-8.472
+0.117
+Neptune
+99.728
+99.997
+4558.857
+-12.377
+-0.109
+Pluto
+99.670
+99.998
+7304.326
+-24.083
+-0.12907
+Table B3: Absolute deviations, with estimated α, of rmax from observation, evaluated in (106
+km).
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmin
+rnum
+min − Obs
+ranal
+min − Obs
+MERCURY
+100.062
+100.006
+46.000
+0.028
+0.002
+Venus
+99.838
+99.994
+107.480
+-0.173
+-0.005
+EARTH
+99.856
+100.003
+147.095
+-0.211
+0.004
+MARS
+99.962
+99.989
+206.650
+-0.077
+-0.022
+JUPITER
+99.897
+99.996
+740.595
+-0.757
+-0.027
+SATURN
+99.802
+100.001
+1357.554
+-2.684
+0.020
+URANUS
+99.905
+99.995
+2732.696
+-2.579
+-0.117
+Neptune
+99.820
+100.002
+4471.050
+-8.037
+0.107
+Pluto
+100.107
+100.002
+4434.987
+4.753
+0.125
+Table B4: Absolute deviations, with estimated α, of rmin from observation, evaluated in (106
+km).
+27
+
+References
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+ph.HE].
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+(2017), arXiv:1706.09979 [physics.atom-ph].
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+29
+

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+arXiv:2301.13533v1  [eess.SY]  31 Jan 2023
+1
+Passivity-based power sharing and voltage regulation
+in DC microgrids with unactuated buses
+Albertus Johannes Malan, Pol Jané-Soniera, Felix Strehle, and Sören Hohmann
+Abstract—In this paper, we propose a novel four-
+stage distributed controller for a DC microgrid that
+achieves power sharing and average voltage regulation
+for the voltages at actuated and unactuated buses. The
+controller is presented for a DC microgrid compris-
+ing multiple distributed generating units (DGUs) with
+time-varying actuation states; dynamic RLC lines; non-
+linear constant impedance, current and power (ZIP)
+loads and a time-varying network topology. The con-
+troller comprising a nonlinear gain, PI controllers, and
+two dynamic distributed averaging stages is designed
+for asymptotic stability. This constitutes first deriving
+passivity properties for the DC microgrid, along with
+each of the controller subsystems. Thereafter, design
+parameters are found through a passivity-based optim-
+isation using the worst-case subsystem properties. The
+resulting closed-loop is robust against DGU actuation
+changes, network topology changes, and microgrid
+parameter changes. The stability and robustness of the
+proposed control is verified via simulations.
+Index Terms—DC microgrids, distributed control,
+passivity, power sharing, voltage regulation.
+I. Introduction
+T
+HE ADVENT of localised power generation and stor-
+age increasingly challenges the prevailing centralised
+power-generation structures. Originally proposed in [1],
+the microgrids paradigm envisions networks that can oper-
+ate autonomously through advanced control while meeting
+consumer requirements. Although current electrical grids
+predominantly use AC, high and low voltage DC networks
+have been made technically feasible due to the continual
+improvements of power electronics. Indeed, DC microgrids
+exhibit significant advantages over their AC counterparts,
+demonstrating a higher efficiency and power quality while
+simultaneously being simpler to regulate [2], [3].
+In microgrids, power generation and storage units
+are typically grouped into distributed generation units
+(DGUs) which connect to the microgrid through a single
+DC-DC converter for higher efficiency [2]. This changes
+the traditionally centralised regulation problem in power
+grids into a problem of coordinating the DGU connected
+throughout the microgrid. This coordination is generally
+This work was supported in part by Germany’s Federal Ministry
+for Economic Affairs and Climate Action (BMWK) through the
+RegEnZell project (reference number 0350062C). (Corresponding
+author: A. J. Malan.)
+A. J. Malan, P. Jané-Soniera, F. Strehle, and S. Hohmann
+are with the Institute of Control Systems (IRS), Karlsruhe In-
+stitute of Technology (KIT), 76131, Karlsruhe, Germany. Emails:
+[email protected], [email protected], [email protected],
+[email protected].
+realised as average or global voltage regulation in combina-
+tion with load sharing between the DGUs (see e.g. [4]–[6]).
+Literature Review: A vast number of approaches have
+been proposed for the voltage regulation and load sharing
+of DC microgrids, as detailed in the overview papers [3],
+[7], [8] along with the sources therein. These approaches
+are broadly categorised as either centralised, decentralised
+or distributed in nature [3], [7], [8]. While centralised
+controllers can optimally coordinate the DGUs, they offer
+reduced scalability and flexibility and have a single point
+of failure [8]. On the other hand, decentralised controllers
+either only attempt to achieve voltage stability [9]–[11]
+or achieve load sharing at the cost of voltage regulation
+quality (e.g. the droop-based approaches in [3]).
+In response to these limitations, numerous controllers
+for voltage regulation and load sharing which operate in a
+distributed manner have been proposed [4]–[6], [12]–[20].
+In [4], distributed averaging is employed to find a global
+voltage estimate with which voltage regulation is achieved,
+but the microgrid dynamics are neglected in the stability
+analysis. Distributed averaging with dynamic microgrid
+models is used in [5], [12], although [5] requires LMIs to
+be solved before buses are allowed to connect whereas
+[12] only considers constant current loads. Similarly, a
+sliding-mode controller is proposed in [13] for a dynamic
+microgrid with constant current loads. On the other hand,
+[14] proposes a cyberattack-resilient controller for a mi-
+crogrid with constant conductance loads and resistive
+lines. A consensus-based distributed controller with event-
+triggered communication is presented in [15]. Consensus-
+based controllers are also utilised in [6], [16], [17], where
+[6] uses a consensus-based integral layer on top of a droop-
+based controller. Finally, while many contributions strive
+to achieve proportional current sharing [4]–[6], [12]–[17],
+[20], nonlinear controllers that achieve proportional power
+sharing have also been proposed in [18], [19].
+While the literature listed above differ greatly in their
+approaches, we note a commonality in their omission of
+buses without actuation. This omission is typically mo-
+tivated either by considering a microgrid comprising only
+actuated DGU buses [4], [5], [16], [17], or by eliminating the
+unactuated buses with the Kron-reduction [6], [12]–[15],
+[18]–[20]. However, considering a network comprising only
+actuated buses severely limits the flexibility of a microgrid,
+since each bus must be able to supply or consume enough
+power at all times. On the other hand, the Kron-reduction
+requires loads to be described as positive conductances
+(see e.g. [21]). While research into Kron-reduced networks
+with negative loads is ongoing (see e.g. [22]), the general
+
+2
+inclusion of negative loads, e.g. non-controllable power
+sources, in Kron-reducible networks remains out of reach
+at present. Furthermore, consider the case where a DGU
+can no longer supply or consume the required amount of
+power, e.g. a fully charged or discharged battery storage.
+Such a DGU then loses the ability to regulate itself and
+fully support the grid. In the approaches considered above
+[4]–[6], [12]–[20], such a DGU is forced to disconnect from
+the microgrid and its local measurements are discarded.
+For DGUs with intermittent power sources, this could
+result in significant swings in the number of controlled and
+observed buses in the microgrid.
+Main Contribution:
+In this paper, we consider a
+DC microgrid as a physically interconnected multi-agent
+system. Extending our work in [23]1, we propose a four-
+stage controller that achieves voltage regulation and power
+sharing in a DC microgrid with actuated and unactuated
+buses in a distributed manner. The four-stage controller
+comprises a nonlinear weighting function, two dynamic
+distributed averaging (DDA) stages and a proportional-
+integral (PI) controller. The asymptotic stability of the
+closed loop comprising the DC microgrid and the four-
+stage controller interconnected in feedback is proven by
+means of passivity theory. In detail, the contributions
+comprise:
+1) A four-stage distributed controller for DC microgrids
+which achieves consensus on the weighted average
+voltage error of actuated and unactuated buses and
+assures coordination through power sharing at the
+actuated buses.
+2) A nonlinear weighting function that penalises voltage
+errors outside a given tolerance band more strongly
+than those within.
+3) Passivity classifications for each of the constitutive
+microgrid subsystems (DGUs, loads, and lines) and
+for each of the controller stages (weighting function,
+DDA, and PI).
+4) A
+method
+for
+calculating
+the
+input-feedforward
+output-feedback passive (IF-OFP) indices of the non-
+linear power-controlled DGUs through optimisation.
+5) An IF-OFP formulation for the DC microgrid with
+a supply rate that is independent of the network
+topology, the number of buses and their states of
+actuation.
+6) A passivity-based stability analysis for the equilib-
+rium of the DC microgrid connected in feedback with
+the four-stage controller.
+In addition to the contributions listed above, we also
+contribute a theoretical result comprising a formalisation
+of the obstacle presented by cascaded input-feedforward
+passive (IFP) and output-feedback passive (OFP) systems
+in the analysis of dissipative systems. This theoretical
+1The controller proposed in [23] is extended by weighing the
+error with a nonlinear function. Moreover, in addition to applying
+the controller to a DC microgrid context, we here propose a new
+dissipativity-based analysis that investigates the closed loop stability
+analytically as opposed to the numerical results in [23].
+contribution informs and motivates parameter choices for
+the four-stage controller in Contribution 1.
+We highlight that the proposed controller can achieve
+exact voltage regulation and power sharing with the
+stability verified with the eigenvalues of the linearised
+system. Moreover, by employing leaky PI controllers, we
+demonstrate a passivity-based stability analysis that is
+independent of and robust against changes in the commu-
+nication topology, changes in the electrical topology, load
+changes, changes in the actuation status of DGUs, uncer-
+tainties in component parameters, and buses connecting
+or disconnecting.
+Paper Organisation: The introduction concludes with
+some notation and preliminaries on graph theory. In
+Section II, we recall and introduce results relating to
+dissipativity theory. Next, in Section III, the problem is
+modelled and objectives for the steady state are formal-
+ised. In Section IV, a four-stage control structure is intro-
+duced that fulfils objectives from Section III. Thereafter,
+the passivity properties of the constituent subsystems are
+investigated in Section V and the controller is designed
+for asymptotic stability of the closed loop in Section VI.
+Finally, in Section VII, a simulation is used to verify the
+asymptotic stability and robustness of the closed loop.
+Concluding remarks are provided in Section VIII.
+Notation and Preliminaries: Define as a vector a =
+(ak) and a matrix A = (akl). 1k is a k-dimensional vector
+of ones and Ik is the identity matrix of dimension k.
+Diag[·] creates a (block-)diagonal matrix from the supplied
+vectors (or matrices). The upper and lower limits of a value
+a are given by a and a. For a variable x, we denote its
+unknown steady state as ˆx, its error state as ˜x := x − ˆx,
+and a desired setpoint as x∗. Whenever clear from context,
+we omit the time dependence of variables.
+We denote by G = (N, E) a finite, weighted, undirected
+graph with vertices N and edges E ⊆ N × N. Let |N| be
+the cardinality of the set N. Let L be the Laplacian matrix
+of G. By arbitrarily assigning directions to each edge in E,
+the incidence matrix E ∈ R|N|×|E| of G is defined by
+ekl =
+
+
+
++1
+if vertex k is the sink of edge l,
+−1
+if vertex k is the source of edge l,
+0
+otherwise.
+(1)
+II. Dissipativity Preliminaries
+We here recall and introduce preliminaries of dissip-
+ativity theory for nonlinear systems. In Section II-A we
+provide definitions relating to dissipativity and passiv-
+ity theory. Thereafter in Section II-B, we investigate
+the passivity properties of static functions. Finally, in
+Section II-C, we recall a result on the interconnection
+of dissipative systems with quadratic supply rates and
+formalise a new result on the limitations of such an
+interconnection.
+
+3
+A. Dissipative Systems
+Consider a nonlinear system
+�
+˙x = f(x, u),
+y = h(x),
+(2)
+where x ∈ Rn, u ∈ Rm, y ∈ Rm and where f : Rn×Rm →
+Rn and h : Rn × Rm → Rm are class C1 functions.
+Definition 1 (Dissipative system, c.f. [24]–[26]). A system
+(2) with a class C1 storage function S : Rn × Rm → R+ is
+dissipative w.r.t. a supply rate w(u, y) if ˙S ≤ w(u, y).
+Definition 2 (Quadratic supply rates, c.f. [24]–[26]). A
+system (2) that is dissipative w.r.t. w(u, y) is
+• passive if w = uTy,
+• input-feedforward passive (IFP) if w = uTy − νuT u,
+• output-feedback passive (OFP) if w = uT y − ρyT y,
+• input-feedforward output-feedback passive (IF-OFP) if
+w = (1 + νρ)uT y − νuT u − ρyT y,
+• has an L2-gain of γL2 if w = γ2
+L2uTu − yT y,
+where γL2 > 0 and ν, ρ ∈ R.
+Definition 3 (Zero-state observable (ZSO) [24, p. 46]). A
+system (2) is ZSO if u ≡ 0 and y ≡ 0 implies x ≡ 0.
+For cases where the desired equilibrium of a system is
+not at the origin but at some constant value, the shifted
+passivity [24, p. 96] or equilibrium-independent passivity
+(EIP) [27] of a system must be investigated. Naturally, this
+requires that an equilibrium exists, i.e. there is a unique
+input ˆu ∈ Rm for every equilibrium ˆx ∈ ˆ
+X ⊂ Rn such that
+(2) produces f(ˆx, ˆu) = 0 and ˆy = h(ˆx, ˆu) [28, p. 24].
+Definition 4 (EIP [28, p. 24]). A system (2) is EIP
+if there exists a class C1 storage function S(x, ˆx, u),
+S : Rn × ˆ
+X × Rm → R+, with S(ˆx, ˆx, ˆu) = 0, that is dis-
+sipative w.r.t. w(u − ˆu, y − ˆy) for any equilibrium (ˆu, ˆy).
+B. Passive Static Functions
+Recall that a sector-bounded static nonlinear function
+is dissipative to a supply rate defined by the sector bound
+[26, Def. 6.2]. We now consider the arbitrarily shifted
+single-input single-output function
+�
+y = h(u),
+u, ˆu ∈ U,
+y, ˆy ∈ Y,
+h : U → Y,
+˜y = ˜h(˜u) := h(u) − h(ˆu) = y − ˆy,
+˜u := u − ˆu
+(3)
+and show how its dissipativity properties may be derived.
+Proposition 5 (EIP static functions). A static function
+(3) of class C0 is IF-OFP(c, 1/c) w.r.t. the arbitrarily
+shifted input-output pair (˜u, ˜y) if
+c ≤ dh(u)
+du
+≤ c,
+∀u ∈ U.
+(4)
+and 0 < c < ∞.
+Proof. Consider for (3) the slope between an arbitrary
+shift (ˆu, ˆy) ∈ U ×Y and a point (u, y), for which the upper
+and lower bounds are given by
+c ≤ y − ˆy
+u − ˆu ≤ c,
+∀(u, y), (ˆu, ˆy) ∈ U × Y.
+(5)
+Changing to the shifted variables ˜u and ˜y as in (5) and
+multiplying through by ˜u2 yields
+c˜u2 ≤ ˜u˜y ≤ c˜u2 ⇐⇒ (˜y − c˜u)(˜y − c˜u) ≤ 0
+⇐⇒ (˜y − c˜u)(1
+c ˜y − ˜u) ≤ 0,
+(6)
+for c > 0, which describes an IF-OFP function (see [26,
+p. 231]). Finally, through the mean value theorem, the
+bounds in (5) may be found from (4).
+■
+We note that the restrictions on c in Prop. 5 are needed
+from a computational point of view (c < ∞) and to ensure
+that the passivity indices correspond to the correct sector2
+(c > 0). However, this limits the passivity properties
+attainable through Prop. 5 to ρ = 1/c > 0.
+Remark 1 (Symmetrical sectors). Placing the additional
+restriction c = −c in (4) results in the Lipschitz continuity
+of h(u). Moreover, this implies that the arbitrarily shifted
+function ˜h(˜u) has a finite L2-gain of c [29].
+C. Interconnected Quadratic Dissipative Systems
+Building upon the results on the interconnection of
+dissipative systems in [28], [30], we now provide a method
+for finding dissipativity properties for a subset of the inter-
+connected subsystems such that interconnected stability is
+guaranteed. Specifically, we look for the dissipative supply
+rates that restrict the subset of subsystems as little as pos-
+sible. For a set S of subsystems, define u = [uT
+1 , . . . , uT
+|S|]T
+and y = [yT
+1 , . . . , yT
+|S|]T .
+Theorem 6 (Minimally restrictive stabilising indices).
+Consider |S| subsystems of the form (2) which are dissipat-
+ive w.r.t. the supply rates wi = 2σiuT
+i yi−νiuT
+i ui−ρiyT
+i yi
+and are linearly interconnected according to u = Hy. The
+stability of the interconnected system is guaranteed if there
+exists a D and νj, ρj ∈ R with j ∈ J such that
+min
+D, νj, ρj,
+j∈J
+�
+j∈J
+(νj + ρj)
+s.t.
+σj = 1/2(1 + νjρj),
+j ∈ J,
+Q ≼ 0,
+D2 ≻ 0
+(7)
+where the subsystems with configurable supply rates are
+represented by the set J ⊂ S, and
+Q :=
+�H
+I
+�T
+DWD
+�H
+I
+�
+(8)
+D := Diag[dT , dT ],
+d = (
+�
+di),
+(9)
+W :=
+�− Diag[νi]
+Diag[σi]
+Diag[σi]
+− Diag[ρi]
+�
+,
+i ∈ S.
+(10)
+2Consider e.g. the sector Prop. 5 would yield if c ≤ c < 0.
+
+4
+The proof for Theorem 6 follows analogously to the
+proof of [29, Theorem 13] with application of [29, Re-
+mark 5] and is thus omitted for brevity. Note that if J = ∅
+in (7), Theorem 6 can be used to verify the stability of
+interconnected dissipative systems.
+Despite the design flexibility provided by Theorem 6,
+certain cascade configurations present obstacles to the ap-
+plication of dissipativity theory. The following proposition
+formalises the problem presented by one such configura-
+tion which arises in the sequel and is used to inform the
+control design.
+Proposition 7 (Non-dissipativity of cascaded IFP-OFP
+systems). Consider |S| ≥ 2 subsystems (2) which are
+dissipative w.r.t. wi = 2σiuT
+i yi − νiuT
+i ui − ρiyT
+i yi and
+linearly interconnected according to u = Hy. Let i = 1 and
+i = 2 arbitrarily denote subsystems that are IFP and OFP,
+respectively. If these systems are connected in exclusive
+casade and do not form a feedback connection, i.e.
+H =
+
+
+0
+0
+∗
+1
+0
+0
+0
+∗
+∗
+
+ ,
+(11)
+then investigating stability via separable storage functions
+as in Theorem 6 fails.
+Proof. Evaluating the stability criteria in (7) under the
+imposed IFP and OFP conditions yields the Q (8) entries
+q11 = d1ρ1 + d2ν2 = 0,
+q12 = q21 = d2σ2
+2
+= d2
+2 . (12)
+Since di > 0, Q constitutes an indefinite saddle-point mat-
+rix [31, Section 3.4], violating the requirement in (7).
+■
+Remark
+2
+(Non-separable
+storage
+functions).
+The
+obstacle in Prop. 7 arises due to the storage functions being
+compartmentalised by the subsystem boundaries. While the
+separability of storage functions is a central motivation for
+the use of dissipativity theory, forgoing this allows for a
+stability analysis through less conservative methods (e.g.
+the KYP lemma).
+III. Problem Description
+In this section, the components comprising the DC mir-
+crogrid are introduced in Section III-A. This is followed by
+Section III-B, where controllers are added which regulate
+the output power of actuated buses in order to facilitate
+power sharing in the sequel. Finally, we formulate the
+coordination and cooperation goals as a control problem
+in Section III-C.
+A. DC Network
+We consider a DC microgrid comprising N = |N| buses
+connected by via π-model electrical lines, as depicted in
+Fig. 1. Let the graph GP = (N, EP) describe the intercon-
+nection with N as the set of buses and EP as the set of
+lines. Without loss of generalisation, we allow each node to
+inject power through a DC-DC buck converter connected
+via a lossy LC-filter. Note that a time-averaged model (see
+e.g. [12]) is used for the buck converter and the energy
+source is assumed to be ideal but finite.
+Let the buses be split into an actuated set Nα and
+an unactuated set Nβ, according to whether the buck
+converter can freely regulate the amount of power injected
+at a given time. Buses may freely switch between the sets
+Nα and Nβ, but Nα ∩ Nβ = ∅ and Nα ∪ Nβ = N always
+hold. To characterise this actuation state of a bus, define
+the piecewise-constant, time-varying actuation parameter
+αk(t) as
+αk(t) :=
+� 1,
+k ∈ Nα,
+0,
+k ∈ Nβ.
+(13)
+Note that we omit the time dependence of αk in the sequel.
+The dynamics for actuated buses with DGUs, where
+αk = 1 with k ∈ Nα are described by
+�
+Lk˙ik
+Ceq,k ˙vk
+�
+=
+�−Rk
+−1
+1
+0
+��ik
+vk
+�
++
+�
+vs,k
+−eT
+P,kit − IL,k(vk)
+�
+(14)
+where Ceq,k = Ck + 1/2eT
+P,k Diag[Ckl]eP,k; Ck, Ckl, Lk > 0;
+ik ∈ R; and vk ∈ R+. The line currents it connect to the
+capacitor voltages according to incidence matrix EP =
+(eT
+P,k) of GP. The dynamics of the unactuated load buses
+with αk = 0 correspond to the simplified system
+Ceq,k ˙vk = −eT
+P,kit − IL,k(vk),
+k ∈ Nβ
+(15)
+In both the actuated (14) and unactuated (15) cases, the
+loads are considered static, nonlinear voltage-dependent
+current sources which are described by class C0 functions.
+In this work, we utilise the standard ZIP-model comprising
+constant impedance, constant current and constant power
+parts. Note that other continuous functions may also be
+used without restriction3. As described in [33, pp. 110–
+112], we define a critical voltage vcrit, typically set to
+0.7vRef, below which the loads are purely resistive. Thus,
+IL,k(vk) =
+
+
+
+Z−1
+k
+· vk + Ik + Pk
+vk
+,
+vk ≥ vcrit,
+Z−1
+crit,k · vk,
+vk < vcrit,
+(16)
+Z−1
+crit,k := IL,k(vcrit)
+vcrit
+= Z−1
+k
++ Ik
+vcrit
++ Pk
+v2
+crit
+,
+(17)
+describes a static, nonlinear load which conforms to (3).
+Lastly, the π-model transmission lines physically con-
+necting the nodes are governed by the dynamics
+Lkl˙it,kl = −Rklit,kl + eT
+P,klv,
+kl ∈ EP,
+(18)
+where it,kl ∈ R, Lkl, Rkl > 0 and (eT
+P,kl)T = EP. Note
+that the line capacitances are included in the equivalent
+capacitances Ceq,k at the buses.
+B. DGU Power Regulator
+To allow for power sharing between the actuated buses
+(14) in the sequel, we equip each DGU with a controller
+3This includes exponential loads (see e.g. [32]).
+
+5
+Linekl
+αk ∈ {0, 1}
+p∗
+k
+vk
+−
++
+vs,k
+ik
+Rk
+Lk
+Ck
+IL,k(vk)
++
+−
+vk
+Buckk
+Busk
+Ckl
+2
+Rkl
+it,kl
+Lkl
+Ckl
+2
+Busl
+Figure 1: Circuit diagram of a bus comprising a DC-DC buck converter, a filter, and a current source representing a
+load, connected to a π-model line (blue); the line capacitances considered to be part of the respective buses.
+that can regulate the injected power to a desired setpoint
+p∗
+k. This regulator has the form
+˙ed,k = αk(p∗
+k − pk)
+vs,k = kP
+d (p∗
+k − pk) + kI
+ded,k + ˜Rkik + vRef
+(19)
+where ed ∈ R, pk = vkik is the actual power injected,
+˜R ∈ R is the damping added to the system, and kP
+d , kI
+d >
+0 are the control parameters. Combining (19) with (14)
+yields the nonlinear system describing the actuated agents
+k ∈ Nα
+
+
+˙ed,k
+Lk˙ik
+Ck ˙vk
+
+=
+
+
+0
+−vk
+0
+kI
+d
+˜Rk − Rk − kP
+d vk
+−1
+0
+1
+0
+
+
+
+
+ed,k
+ik
+vk
+
+
++
+
+
+p∗
+k
+kP
+d p∗
+k + vRef
+−eT
+k it − IL,k(vk)
+
+
+(20)
+Remark 3 (Regulating current or voltage). Without in-
+validating the stability analysis in the sequel, the regulator
+in (19) can be exchanged for simpler, purely linear current
+or voltage regulators (see e.g. [9]–[11]).
+Remark 4 (Constrained DGU operation). If an actuated
+DGU cannot provide the desired power p∗
+k, e.g. due to
+current, storage or temperature limitations, the DGU may
+simply set its actuation state αk = 0 to disable its control. If
+some power can still be supplied, it may simply be regarded
+as a negative load. This allows DGUs to contribute to the
+power supply of the network, even in the face of control
+limitations.
+C. Control Problem
+A central requirement for DC microgrids is voltage
+stability, which requires the bus voltages to remain within
+a given tolerance band around the reference vRef. Spe-
+cifically, this requirement should be met throughout the
+network, and not only at the actuated buses. Due to
+the presence of lossy lines, power flows are associated
+with voltage differences between buses, meaning that
+vk → vRef, ∀k ∈ N is not practical. Ideally, the voltages at
+all buses should be arrayed in the tolerance band around
+vRef and be as close to vRef as possible4. The manipulated
+variables used to achieve this are the power setpoints p∗
+k
+supplied to the actuated DGUs (19). This leads to the
+first objective for the control of the DC microgrid, which
+involves finding the setpoints p∗
+k that ensure the weighted
+average voltage equals vRef at steady state.
+Objective 1 (Weighted voltage consensus).
+Find p∗
+k s.t. lim
+t→∞
+1
+N
+�
+k∈N
+h(vk(t)) = vRef
+(21)
+for a strictly increasing weighting function h : R → R.
+By choosing a nonlinear h, large voltage errors may be
+weighed more strongly. This allows for better utilisation of
+the tolerance band since bus voltages can be further from
+vRef before registering as a significant error.
+In addition to Objective 1, it is desired that all actuated
+DGUs contribute towards supplying and stabilising this
+network. Ensuring that all DGUs receive the same setpoint
+spreads the load across actuated buses, leading to a reduc-
+tion in localised stress on the DGUs. We thus formulate
+the second objective as requiring uniform setpoints for the
+DGUs in steady state.
+Objective 2 (Cooperative power sharing).
+lim
+t→∞(p∗
+k(t) − p∗
+l (t)) = 0,
+∀ k, l ∈ N
+(22)
+Achieving Objectives 1 and 2 thus yields a controlled
+microgrid where the average weighted voltage error of all
+buses tends to zero through the coordinated action of the
+actuated buses in a distributed fashion. These objectives
+also allow DGUs to transition seamlessly between actuated
+and unactated states and ensure no measurement inform-
+ation is discarded simply because a bus cannot regulate
+itself. Notice that disregarding the unactuated buses in
+Objectives 1 and 2 yields the objectives typically used in
+the literature [4], [6], [12]–[14], [16], [17], [20].
+To achieve these objectives, we make the following
+assumptions related to appropriate network design.
+Assumption 1 (Feasible network). The available power
+sources can feasibly supply the loads with power over the
+4The magnitude of the errors vRef − vk strongly depend on the
+loads and line resistance. Small errors therefore presuppose adequate
+network design.
+
+6
+hw
+hw
+DDA2,1
+DDA2,N
+PI1
+PIN
+DDA4,1
+DDA4,N
+DC MG
+Stage 1
+Stage 2
+Stage 3
+Stage 4
+uw
+uw,1
+uw,N
+yw,1
+yw,N
+ya,2,1
+ya,2,N
+yc,1
+yc,N
+ya,4,1
+ya,4,N
+p∗
+v
+−
+vRef1N
++
+Figure 2: Distributed four-stage control connected in feed-
+back to the microgrid and with indicated communication
+links
+between the local control structures.
+given electrical network, i.e. a suitable equilibrium for the
+microgrid exists.
+Assumption 2 (Number of actuated DGUs). At least one
+DGU is actuated at any given time, i.e. Nα ̸= ∅.
+Assumption 3 (Connected topologies). Objectives 1
+and 2 only apply to a subset of buses electrically connected
+as per GP. Moreover, for a distributed control, a connected
+communication graph exclusively interconnects the same
+subset of buses.
+Note that Assumption 1 is a typically made implicitly or
+explicitly in the literature (see e.g. the discussion in [16]).
+Assumptions 2 and 3 further specify requirements that
+allow a distributed control to achieve the feasible state
+in Assumption 1, i.e. by ensuring that at least one source
+of stabilisation is present in the network (Assumption 3),
+and by ensuring that the coordination corresponds to the
+network to be controlled Remark 5.
+Remark 5 (Proportional power sharing). By normalising
+the power setpoint p∗
+k and weighing the input in (19)
+according to the rated power of a given DGU, Objective 2
+automatically describes a proportional power sharing. With
+reference to Remark 4, this also allows the constrained
+DGUs to lower their maximum injectable power instead
+of setting the DGUs to the unactuated state αk = 0. We
+omit the extension to proportional power sharing in this
+work for simplicity.
+IV. Control Structure
+To meet Objectives 1 and 2, we propose the four-
+stage control structure depicted in Fig. 2. This control
+structure comprises two DDA implementations separated
+by agent PI controllers local to the buses as in [23]. This
+is prepended by a nonlinear weighting function hw. In the
+Sections IV-A, IV-B and IV-C, we successively introduce
+these respective subsystems. Finally in Section IV-D, we
+show that the control structure meets the objectives.
+A. DDA Controller
+Consider the communiation graph GC = (N, EC) linking
+the buses of the DC microgrid. The communication graph
+comprises the same vertices as the physical interconnection
+graph GP but possibly with a different topology. Let LC
+denote the Laplacian of GC. For Stages 2 and 4 of the
+control structure, each agent implements an instance of
+the DDA5 described in [34]. The instances of the respective
+stages may be combined into vector form as
+DDAs
+
+
+
+
+
+� ˙xa,s
+˙za,s
+�
+=
+�
+−γaIN −LC,P
+LT
+C,I
+−LC,I
+0
+��
+xa,s
+za,s
+�
++
+�
+γaIN
+0
+�
+ua,s,
+ya,s = xa,s,
+(23)
+where s
+∈
+{2, 4} denotes the stage in Fig. 2, and
+xa,s, za,s ∈ RN are the consensus and integral states
+respectively. Furthermore, γa > 0 is a global estimator
+parameter (see [34]), and LC,I = kI
+aLC and LC,P = kP
+a LC
+are Laplacian matrices weighted for the integral and pro-
+portional responses, respectively. Recall from [34] that a
+constant input ua,s yields
+lim
+t→∞ ya,s,k = uT
+a,s1N
+N
+,
+∀ k.
+(24)
+B. Agent PI Controller
+In Stage 3, we equip each bus k ∈ N with a leaky agent
+PI controller similar to the approach in [35]
+PIk
+�
+˙xc,k = −ζcxc,k + uc,k,
+yc,k = kI
+cxc,k + kP
+c uc,k,
+(25)
+where xc,k ∈ R, ζc ≥ 0 and kP
+c , kI
+c > 0. Note that ζc = 0
+reduces (25) to an ideal PI controller. The combined form
+of the N agent controllers is
+˙xc = −ζcxc + uc,
+yc = kI
+cxc + kP
+c uc
+(26)
+Remark 6 (Non-ideal integrators). As shown in the
+sequel, ideal PI controllers only exhibit an IFP property,
+whereas the DDA controller is OFP. The interconnection
+in Fig. 2 thus yields a cascaded IFP-OFP structure which
+obstructs the dissipativity analysis (see Prop. 7). The use
+of leaky integrators (ζc > 0) overcomes this obstacle at the
+cost of negatively affecting the steady-state properties, since
+(25) forces the equilibrium
+uc = ζcxc
+(27)
+instead of uc = 0. In the context of Fig. 2, this corresponds
+to a unwanted steady-state offset for the average weighted
+voltage error.
+Remark 7 (Agent PI controller anti-windup). To prevent
+controller windup, the input to the PI control in (25) should
+be zeroed for any unactuated agents that are disconnected
+from the communication network.
+Remark
+8
+(Non-participating agents). Implementing
+(25) at each bus k ∈ N allows for a faster reaction to
+disturbances at the cost of controller redundancy. By setting
+ua,4,m := ya,4,m at Stage 4 DDA of the control structure
+5We implement the PI-DDA variant proposed in [34] and use the
+same communication graph for the proportional and integral terms.
+
+7
+u
+y
+hw(u)
+dhw(u)
+duw
+Figure 3: Example of the weighting function hw (28) and
+its derivative (58) on a unit grid, with aw = 0.5, bw = 1.5
+and cw = 2.
+for some agents m ∈ M ⊂ N, the PI control (25)
+can be omitted at the agents in M without affecting the
+steady state. Nevertheless, the measurements of the buses
+in k ∈ M are still included in the Stage 2 DDA. Note that
+at least one participating agent PI controller is required
+(see [23, Remark 8]).
+C. Weighting Function
+To allow for a better utilisation of the tolerance band
+around vRef, we desire a weighting function that assigns a
+low gain for errors within the tolerance band and a high
+gain for larger errors. We therefore define the class C1
+function yw,k = hw(uw,k) conforming to (3), where
+hw(u) := awu + bwgw(u) − bw tanh(gw(u)),
+(28)
+gw(u) :=
+
+
+
+u + cw,
+u < −cw
+0,
+−cw ≤ u ≤ cw
+u − cw,
+cw < u
+(29)
+and where (29) describes a dead-zone parametrised by
+cw. An example of (28) is depicted in Fig. 3 along with
+its derivative. For a strictly increasing function as per
+Objective 1, set aw > 0 and bw > −aw.
+D. Equilibrium Analysis
+In a first step towards analysing the closed loop, we
+analyse the assumed equilibrium of the interconnected
+microgrid and four-stage controller (see Assumption 1).
+Specifically, we verify that the proposed control yields an
+equilibrium which satisfies Objectives 1 and 2.
+Proposition 8 (Controller equilibrium analysis). Con-
+sider the DC microgrid comprising (15), (18), and (20)
+which is connected in feedback with the four-stage con-
+troller comprising (23), (26), and (28) as in Fig. 2. Let
+Assumptions 1, 2 and 3 hold. Then, Objective 2 is met for
+the equilibrium imposed by the control structure. Moreover,
+Objective 1 is achieved exactly for ideal integrators ζc = 0
+in (26). For lossy integrators with ζc > 0, the remaining
+error for Objective 1 is be described by the steady-state
+value of ya,2, where
+ya,2 =
+ζc
+kIc(1 + ζckPc )ya,4.
+(30)
+The proof of Prop. 8 can be found in Appendix A.
+Through Prop. 8 we thus confirm that the proposed con-
+troller yields an equilibrium which meets the requirements,
+even though the requirements are not perfectly met when
+leaky agent PI controllers are used. We also note that
+Prop. 8 only considers the controlled microgrid already in
+equilibrium and does not consider the convergence to the
+equilibrium.
+Remark 9 (Compensating leaky-integral errors). As in-
+dicated by (30) in Prop. 8, the leaky agent PI controllers
+result in a constant steady-state error for the average
+voltage regulation (Objective 1). Since a positive ya,2 cor-
+responds to voltages below the desired vRef, it follows
+that setting vRef above the actual desired voltage reference
+will result in higher bus voltages. Changing vRef thus
+allows the steady-state effects of the leaky integrators to be
+compensated. Moreover, notice that ya,4 is the controller
+output, i.e. the power setpoint p∗ used for the DGUs
+(see Fig. 2). Thus, the error measure in (30), which is
+only dependent on the controller output, can be used to
+determine the offset to vRef for exact voltage regulation.
+Note, however, that modifying vRef based on p∗ results in
+a new loop which requires an additional stability analysis.
+V. Subsystem Passivity Analysis
+Having verified whether the desirable steady state is
+achieved by the controller, we now set about analysing the
+convergence to this steady state. With the aim of applying
+Theorem 6 for the closed-loop stability, we first analyse the
+passivity properties of the individual subsystems. Since
+the steady-state bus voltages ˆvk are unknown and non-
+zero, we investigate the passivity properties shifted to any
+plausible point of operation using EIP. To this end, we
+construct an EIP formulation for the DC microgrid from
+its constitutive elements in Section V-A. This is followed
+by the respective analyses of the various controller stages
+in Section V-B. Note that we omit the bus indices k and
+l in this section where clear from context.
+A. DC Microgrid Passivity
+For the stability of the microgrid at the equilibrium
+ˆv, we desire an EIP property relating the shifted input
+power setpoints ˜p∗ = p∗ − ˆp∗ to the output voltage errors
+˜v = v − ˆv of all nodes, since this port (˜p∗, ˜v) is used
+by the controller in Fig. 2. To this end, we derive EIP
+properties for the load, DGU and line subsystems of the
+microgrid, making sure to shift the subsystem dynamics to
+the assumed equilibrium in each case (see Assumption 1).
+Thereafter, we combine the results of these subsystems, to
+construct an EIP property for the microgrid as a whole.
+Where applicable, an analysis of the zero-state dynamics is
+performed to ensure the eventual stability of the controlled
+microgrid.
+1) Load Passivity: Let the unactuated bus dynamics
+in (15) for the buses in Nβ be shifted to the equilibrium
+(ˆit, ˆv), yielding
+Ceq ˙˜v = −eT
+P,k˜it − ˜IL(˜v) + (eT
+P,kˆit + IL(ˆv)),
+(31)
+
+8
+for the static load function shifted according to (3). In
+(31), eT
+P,kˆit = −IL(ˆv) since the load is fully supplied by
+the cumulative line currents in steady state.
+Proposition 9 (Load EIP). The shifted load dynamics in
+(31) are OFP(ρL) w.r.t. the input-output pair (−eT
+P,k˜it, ˜v)
+with ρL = cL the smallest gradient of the static load
+function IL(v).
+Proof. Consider the storage function SL along with its
+time derivative
+SL = Ceq
+2 ˜v2,
+(32)
+˙SL = −˜veT
+P,k˜it − ˜v ˜IL(˜v).
+(33)
+Since the static load function IL(v) is IF-OFP according
+to Prop. 5, it is bounded from below by cL˜v2 ≤ ˜v ˜IL(˜v)
+(see (6)). Incorporate this lower bound into (33) to obtain
+˙SL ≤ wL := −˜veT
+P,k˜it − cL˜v2
+(34)
+which yields the OFP property from Definition 2.
+■
+Remark 10 (ZIP load passivity). Prop. 9 and (4) demon-
+strate that the passivity properties of the unactuated buses
+are directly linked to the smallest gradient of the load
+function. For the ZIP load in (16), this yields
+cL = min
+�
+Z−1, Z−1 −
+P
+v2
+crit
+, Z−1
+crit
+�
+.
+(35)
+Considering the strictly passive case (cL = 0) along with
+I, P ≥ 0 yields the passivity condition Z−1v2
+crit ≥ P
+frequently used in the literature [10], [16], [18]–[20].
+2) DGU Passivity: Shift the states (e, i, v) and inputs
+(p∗, it) of the DGU dynamics in (20) for the buses in Nα to
+the respective error variables (˜e,˜i, ˜v) and (˜p∗,˜it) to obtain
+(36) on the next page, where the static load function
+is incorporated into the matrix Ad. Furthermore, the
+measured power p = vi = v(˜i + ˆi) in (19) is left partially
+in unshifted variables such that Ad is also dependent on
+the unshifted voltage v and the steady-state current ˆi.
+Note that the constant τd in (36) is found by setting the
+error variables (˜p∗,˜it, ˜ed,˜i, ˜v) and their time derivatives to
+zero. As such, the constant τd ≡ 0 can be disregarded
+in the passivity analysis. We now analyse the shifted
+nonlinear system in (36) for EIP.
+Theorem 10 (EIP DGUs). The shifted DGU dynamics in
+(36) are simultaneously IF-OFP(νd,1, ρd) w.r.t. the input-
+output pair (˜p∗, ˜v) and IFP(νd,2) w.r.t. the input-output
+pair (−eT
+k ˜it, ˜v), if a feasible solution can be found for
+max
+Pd, νd,1, νd,2, ρd νd,1 + νd,2 + ρd
+s.t.
+(38) holds ∀ v ∈ V ⊆ R+, ∀ˆi ∈ ˆI ⊆ R
+(37)
+where Qd(v,ˆi, cL) := PdAd(v,ˆi, cL) + AT
+d (v,ˆi, cL)Pd,
+Ad(v,ˆi, cL) =
+
+
+0
+−v
+−ˆi
+kI
+d
+˜R − R − kP
+d v
+−1 − kP
+c ˆi
+0
+1
+−cL
+
+ ,
+(39)
+and with νd,1, νd,2, ρd ∈ R, cL as in (4) and cd = [0, 0, 1]T.
+Proof. Consider for (36) the storage function
+Sd =
+
+
+˜ed
+˜i
+˜v
+
+
+T
+Pd
+
+
+˜ed
+L˜i
+Ceq˜v
+
+ ,
+(40)
+with Pd ≻ 0. The time derivative of (40) is
+˙Sd =
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+
+T
+
+Qd(v,ˆi,
+˜IL(˜v)
+˜v
+)
+Pdbd,1
+Pdbd,2
+bT
+d,1Pd
+0
+0
+bT
+d,2Pd
+0
+0
+
+
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+,
+(41)
+with ˜xd as in (36). Since it follows from (6) that −˜v ˜IL(˜v) ≤
+−cL˜v2, this bound can be incorporated into the inequality
+˙Sd ≤
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+
+T
+
+Qd(v,ˆi, cL)
+Pdbd,1
+Pdbd,2
+bT
+d,1Pd
+0
+0
+bT
+d,2Pd
+0
+0
+
+
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+.
+(42)
+The desired IF-OFP and IFP properties for the DGU are
+described by the supply rate
+wd = (1 + νd,1ρd)˜p∗˜v − νd,1(˜p∗)2 − ρd˜v2
+− ˜veT
+P,k˜it − νd,2
+�
+eT
+P,k˜it
+�2
+(43)
+These properties are guaranteed, if ˙Sd−wd < 0 for all valid
+inputs and outputs and for v ∈ V and ˆi ∈ ˆI. Combining
+(42) and (43) in this manner directly leads to constraint
+(38) in (37). Finally, the objective function in (37) seeks
+to find the largest indices for which the constraints are
+satisfied in a similar manner to Theorem 6.
+■
+Although Theorem 10 demonstrates the EIP of the
+actuated buses, notice that the ˜ed and ˆi of (36) are
+not included in the supply rate wd in (43). As such, an
+investigation of the zero state dynamics of the DGU is
+required.
+Proposition 11 (ZSO DGUs). The shifted DGU dynam-
+ics in (36) are ZSO.
+Proof. In (36), set the inputs ˜p∗ ≡ 0, ˜it ≡ 0 and the
+output ˜v ≡ 0. Since τd = 0 and ˜IL(0) = 0, verify from the
+equation for ˙˜v that ˜i ≡ 0. From the equation for ˙˜i, it then
+follows that ˜ed ≡ 0 which concludes this proof.
+■
+Remark
+11
+(Compensating
+non-passive
+loads).
+As
+demonstrated in [11], adding a term dependent on ˙vk to
+the regulator output vs,k in (19) allows for damping to
+be added to the unactuated state vk. This in turn allows
+for regulation in the presence of non-passive loads and
+can yield more favourable passivity indices when applying
+Theorem 10.
+3) Line Passivity: The dynamics of the line subsystem
+(18) shifted to the equilibrium (ˆit, ˆv) yield
+Lkl˙˜it = −Rkl˜it + eT
+P,kl˜v,
+(44)
+which can now be analysed for passivity.
+
+9
+
+
+˙˜ed
+L˙˜i
+Ceq ˙˜v
+
+=
+
+
+0
+−v
+−ˆi
+kI
+d
+˜R−R−kP
+d v
+−1−kP
+c ˆi
+0
+1
+−
+˜IL(˜v)
+˜v
+
+
+�
+��
+�
+Ad(v,ˆi,
+˜IL(˜v)
+˜v
+)
+
+
+˜ed
+˜i
+˜v
+
+
+����
+˜xd
++
+
+
+1
+kP
+d
+0
+
+
+����
+bd,1
+˜p∗ −
+
+
+0
+0
+1
+
+
+����
+bd,2
+eT
+P,k˜it +
+
+
+ˆp∗ − ˆvˆi
+kI
+c ˆed + ( ˜R−R)ˆi − ˆv + vRef − kP
+c (ˆp∗ − ˆvˆi)
+ˆi − eT
+P,kˆit − IL(ˆv)
+
+
+�
+��
+�
+τd
+(36)
+
+
+Qd(v,ˆi, cL) + ρdcdcT
+d
+Pdbd,1 − 1+νd,1ρd
+2
+cd
+Pdbd,2 − 1
+2cd
+bT
+d,1Pd − 1+νd,1ρd
+2
+cT
+d
+νd,1
+0
+bT
+d,2Pd − 1
+2cT
+d
+0
+νd,2
+
+ ≺ 0,
+Pd ≻ 0
+(38)
+Proposition 12 (OFP lines). The shifted line dynamics
+in (44) are OFP(ρt) with ρt = Rkl w.r.t. the input-output
+pair (eT
+P,kl˜v,˜it) with the storage function
+St = Lkl
+2
+˜i2
+t.
+(45)
+Proof. The proof follows trivially by verifying that
+˙St = ˜iteT
+P,kl˜v − Rkl˜i2
+t =: wt,
+(46)
+where wt in an OFP supply rate as per Definition 2.
+■
+4) Interconnected Microgrid Dissipativity: Having sep-
+arately analysed the subsystems comprising the microgrid,
+we now combine the results to formulate the dissipativity
+of the full microgrid w.r.t. the input-output pair (˜p∗, ˜v).
+For simplicity, we group the buses according to their
+actuation states (13). Thus, ˜p∗ = [ ˜p∗
+α
+T , ˜p∗
+β
+T ]T and ˜v =
+[˜vT
+α , ˜vT
+β ]T have the same dimensions. Note that we include
+the inputs ˜p∗
+β for the unactuated buses in Nβ as provided
+by the four-stage controller (see Fig. 2), even though these
+inputs are not used.
+Proposition 13 (Microgrid dissipativity). A DC mi-
+crogrid comprising DGUs (20), lines (18) and loads (15)
+with an interconnection topology described by a connected
+graph GP is dissipative w.r.t. the supply rate
+wM,αβ = (1 + νd,1ρd)˜p∗
+α
+T ˜vα − νd,1 ˜p∗
+α
+T ˜p∗
+α
+− ρd˜vT
+α ˜vα − ρL˜vT
+β ˜vβ,
+(47)
+if νd,2 + ρt ≥ 0 for the worst-case indices of the buses
+and lines calculated in Prop. 9 (ρL), Prop. 12 (ρt), and
+Theorem 10 (νd,1, νd,2, ρd), i.e.
+νd,1 = min
+k∈Nα νd,1,k, νd,2 = min
+k∈Nα νd,2,k, ρd = min
+k∈Nα ρd,k,
+ρL = min
+k∈Nβ ρL,k,
+ρt = min
+kl∈EP ρt,k.
+(48)
+Proof. Define for the interconnected microgrid the storage
+function
+SM =
+�
+k∈Nα
+Sd,k +
+�
+k∈Nβ
+SL,k +
+�
+kl∈EP
+St,kl.
+(49)
+An upper bound for time derivative of (49) may then be
+found by combining the supply rates in (34), (43) and (46)
+˙SM ≤ (1 + νd,1ρd)˜p∗
+α
+T ˜vα − νd,1 ˜p∗
+α
+T ˜p∗
+α − ρd˜vT
+α ˜vα
++ ˜iT
+t ET ˜v − ˜vT
+α Eα˜it − ˜vT
+β Eβ˜it
+− ρL˜vT
+β ˜vβ − (νd,2 + ρt)˜iT
+t ˜it.
+(50)
+The skew-symmetric interconnection of the nodes and lines
+results in ˜iT
+t ET ˜v = ˜vT
+α Eα˜it + ˜vT
+β Eβ˜it. Furthermore with
+νd,2+ρt ≥ 0, we can drop the unnecessary strictly negative
+˜iT
+t ˜it term and verify that ˙SM ≤ wM,αβ.
+■
+Through Prop. 13, the dissipativity of the entire mi-
+crogrid is formulated using the desired input and output
+vectors. However, the supply rate in (47) is dependent on
+the actuation states of the buses. We now remove this
+dependence by finding a supply rate for a specific bus that
+encompasses both its actuated and unactuated state. By
+considering a quadratic supply rate as a sector condition
+(see [26], [29]), a combined supply rate is found through
+the union of the sectors for the actuated and unactuated
+cases.
+Theorem 14 (Actuation independent passivity). A DC
+microgrid for which Prop. 13 holds is IF-OFP(νd,1, ρd)
+w.r.t. the supply rate
+wM = (1 + νd,1ρd)˜p∗T ˜v − νd,1 ˜p∗T ˜p∗ − ρd˜vT ˜v
+(51)
+if, for an arbitrarily small νL > 0,
+0 ≤ νd,2 + ρt,
+(52)
+0 < ρL < 1,
+(53)
+0 > νd,1.
+(54)
+The proof of Theorem 14 can be found in Appendix A.
+Through (51), we thus show that a single IF-OFP supply
+rate describes the input-output passivity of the entire
+microgrid, irrespective of the states of actuation of the
+buses. This supply rate is derived from the properties of
+the DGUs in Theorem 10 and accounts for the worst-case
+loads.
+Remark 12 (Non-passive loads at DGUs). While (53)
+in Theorem 14 requires strictly passive loads at unactuated
+buses, this is not required for the loads at actuated buses.
+
+10
+Indeed, the loads at DGUs may exhibit a lack of passivity
+with cL < 0. However, this would be reflected by the indices
+obtained in Theorem 10 and the supply rate in (51).
+Remark 13 (Non-static loads). Due to the use of passivity
+in this section, the analysis presented here effortlessly
+extends to the case of dynamic loads. Such dynamic loads
+simply need to exhibit equivalent IFP properties (see e.g.
+Prop. 9) and must be ZSO.
+Remark 14 (Passivity-based controllers). In addition
+to the four-stage controller proposed in this work, the
+passivity formulation of the DC microgrid in Theorem 14
+can be used alongside any other controller which provides
+suitable passivity indices. This includes methods such as
+interconnection and damping assignment passivity-based
+control [24, p. 190] or passivity-based model predictive
+control (see e.g. [36]).
+B. Controller Passivity
+Having analysed the passivity of the microgrid subsys-
+tems and their interconnection, we now investigate the
+passivity properties of the control structure in Section IV.
+This is done successively for each part of the controller:
+the DDA stages, the PI stage and the weighting function.
+1) DDA Passivity: Consider the DDA stages in Fig. 2.
+Proposition 15 (DDA Passivity). The DDA controller
+in (23) with the storage function
+Sa,s =
+1
+2γa
+�
+xT
+a,sxa,s + zT
+a,sza,s
+�
+(55)
+is OFP(ρa), ρa = 1, w.r.t. (ua,s, ya,s) and is ZSO.
+Proof. The time derivative of (55) is
+˙Sa,s = −xT
+a,sxa,s − 1
+γa
+xT
+a,sLC,P xa,s + xT
+a,sua,s
+≤ wa,s := xT
+a,sua,s − xT
+a,sxa,s
+(56)
+since LC,P > 0 and γa > 0, thus verifying the OFP
+property for ya,s = xa,s. Furthermore, the DDA controller
+is ZSO since the system dynamics in (23) is Hurwitz [34,
+Theorem 5].
+■
+The OFP result in Prop. 15 also means that (23) has an
+L2-gain of 1 [28, p. 3]. Note that since the DDA in (23) is
+linear, the properties in Prop. 15 also hold for the shifted
+input-output combination (˜ua,s, ˜ya,s) [28, p. 26].
+2) PI Passivity: The ideal PI controller in (26) with
+ζc = 0 can trivially be shown to be IFP(kP
+c ) for the storage
+function Sc = kIcxTc xc/2. The leaky PI control with ζc > 0
+exhibits the following properties.
+Proposition 16 (Leaky PI Passivity). The leaky PI
+control in (26) with the storage function Sc = kIcxTc xc/2
+is dissipative w.r.t. the supply rate
+wc =
+�
+1+2ζckP
+c
+kIc
+�
+�
+��
+�
+2σc
+uT
+c yc −
+�
+kP
+c +ζckP
+c
+2
+kIc
+�
+�
+��
+�
+νc
+uT
+c uc − ζc
+kIc
+����
+ρc
+yT
+c yc
+(57)
+Proof. Calculate the time derivative of Sc as
+˙Sc
+=
+kI
+cxT
+c uc −ζckI
+cxT
+c xc. Substitute in kI
+cxc = yc −kP
+c uc from
+the output in (26) and simplify to verify that ˙Sc = wc.
+■
+Note that while wc in (57) has a quadratic form, it does
+not directly match the IF-OFP form in Definition 2. How-
+ever, by appropriately weighing the storage function Sc,
+the form in Definition 2 is easily obtained. For simplicity
+and without invalidating the results in the sequel, we omit
+this step here. Furthermore, we note that the linearity of
+(26) ensures that the properties in Prop. 16 also hold for
+the shifted input-output combination (˜uc, ˜yc) [28, p. 26].
+3) Weighting Function Passivity: The derivative of the
+weighting function in (28) is described by (see e.g. Fig. 3)
+dyw
+duw
+= aw + bw tanh2(gw(uw)).
+(58)
+By setting bw > −aw and applying Prop. 5, (28) is found
+to be IF-OFP(νw, ρw) with
+νw = aw,
+ρw =
+1
+aw + bw
+,
+(59)
+VI. Interconnected Stability
+Using the passivity properties of the microgrid and
+controller subsystems obtained in Section V, we now in-
+vestigate the stability of the microgrid and controller
+interconnected as in Fig. 2. However, we note that the
+agent PI controller and the Stage 4 DDA controller ex-
+hibit a cascaded IFP-OFP obstacle (see Prop. 7) if the
+PI controller is ideal (ζc = 0) which prevents a closed-
+loop analysis with dissipativity. Thus, in Section VI-A, we
+derive stability conditions using leaky agent PI controllers
+with ζc > 0.
+A. Leaky PI-Controlled Stability
+Consider the case where the passivity properties of all
+subsystems in Fig. 2 except for the weighting function
+(28) are fixed. Combining the results in Section V with
+Theorem 6, we now determine the weighting function
+parameters which guarantee closed-loop stability.
+Theorem
+17
+(Designed
+closed-loop
+stability).
+The
+closed-loop in Fig. 2 is guaranteed to be asymptotically
+stable for the weighting function parameters aw = νw,
+bw = 1/ρw − aw if a feasible solution is found for
+min
+νw, ρw, di,
+νw + ρw
+s.t.
+Q ≺ 0,
+di > 0,
+i = 1, . . . , 5,
+(60)
+where σw = 1/2(1 + νwρw), σd = 1/2(1 + νd,1ρd), and
+Q=
+
+
+−ρwd1
+d2
+2
+0
+0
+−σwd1
+d2
+2
+−ρad2−νcd3 σcd3
+0
+0
+0
+σcd3
+−ρcd3
+kP
+c d4
+2
+0
+0
+0
+kP
+c d4
+2
+−ρad4−νd,1d5
+σdd5
+−σwd1
+0
+0
+σdd5
+−ρdd5−νwd1
+
+
+(61)
+Proof. Use the supply rates for the DC microgrid in (51),
+the two DDA controllers in (56), the agent PI controller
+
+11
+in (57), and the IF-OFP supply rate for the weighting
+function (59) to construct W in (10). Let the output of
+the PI controller be normalised according to
+yc = kI
+cxc + kP
+c uc = kP
+c (κI
+cxc + uc) = kP
+c yκ
+c .
+(62)
+Furthermore, the five subsystems in Fig. 2 are intercon-
+nected by u = Hy, where
+H =
+
+���
+0
+0
+0
+0
+−1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+kP
+c
+0
+0
+0
+0
+0
+1
+0
+
+
+.
+(63)
+Apply Theorem 6, with D as in (9) and simplify Q in (8)
+to obtain (61). This yields the optimisation problem (60),
+where the indices of the weighting function (νw, ρw) are
+configurable. Asymptotic stability is ensured by changing
+the matrix inequality in (7) to a strict inequality and by
+ensuring that any states not present in y are asymptot-
+ically stable. The latter condition is ensured through the
+zero-state analyses in Prop. 11 and Prop. 15 and through
+the condition in Prop. 13. Finally, the parameters aw and
+bw are calculated from (59).
+■
+Through the application of Theorem 17, the parameters
+for the weighting function can thus be designed to ensure
+stability. We highlight that the results in Section V and
+Theorem 17 hold irrespective of the physical or commu-
+nication topologies and are independent of the actuation
+states of the nodes, as long as Assumptions 2 and 3
+hold. Therefore, verifying Theorem 17 ensures robust-
+ness against any changes which do not alter the worst-
+case passivity indices of the respective subsystems (see
+(48)). Note that the presented stability analysis requires
+strictly passive loads and leaky agent PI controllers (see
+Remark 6). As demonstrated via simulation, these require-
+ments are sufficient for stability, but not necessary.
+VII. Simulation
+In this section, we demonstrate the coordination and
+robustness of the proposed control structure by means of
+a Matlab/Simulink simulation using Simscape com-
+ponents. We consider the network comprising 10 buses
+depicted in Fig. 4. In Section VII-A, we describe the setup
+of the simulation along with the various changes that the
+network is subjected to. Next, in Section VII-B, simula-
+tion results are presented for the case where Theorem 17
+holds, i.e. with strictly passive loads and leaky agent PI
+controllers. Finally, in Section VII-C, we show the robust
+stability of the proposed control structure for passive loads
+and ideal agent PI controllers.
+A. Simulation Setup
+The DC microgrid in Fig. 4 is simulated with the
+parameters in Table I. The ZIP load parameters are
+chosen randomly in the specified ranges such that the
+required passivity measures are fulfilled (see Remark 10).
+Table I: Simulation Parameter Values
+Voltages
+vRef = 380 V
+vcrit = 266 V
+DGU Filters (14)
+Rk = 0.2 Ω
+Lk = 1.8 mH
+Ck = 2.2 mF
+ZIP Loads (16)
+|Z−1| ≤ 0.1/Ω
+|I| ≤ 21 A
+|P | ≤ 3 kW
+Elec. Lines (18)
+Rkl = 0.1 Ω/km
+Lkl = 2 µH/km
+Ckl = 22 nF/km
+length ∈ [0.2; 10] km
+Table II: Controller Parameter Values
+Power PI Control (19)
+kP
+d = 90
+kI
+d = 90
+˜R = −8
+DDA Control (23)
+kP
+a = 50
+kI
+a = 100
+γa = 16
+Agent PI Control (26)
+kP
+c = 160
+kI
+c = 600
+ζc = 0.08
+Weighting Function (28)
+aw = 0.1
+bw = 1.1
+cw = 7.5 V
+Furthermore, typical values are used for the DGUs and
+the lines [4], [9], [13]. The lines exhibit the same per
+kilometer parameter values and the line length are chosen
+randomly in the given interval. The line lengths are given
+in Appendix B.
+The simulation starts off in State A (see Fig. 4) with
+Bus 9 connected and with all states at zero. The following
+changes are made at the indicated times.
+• t = 5 s: The actuation states αi of the buses switches
+from State A to State B and Bus 9 is disconnected.
+• t = 10 s: The communication topology switches from
+State A to State B and Bus 10 is connected.
+• t = 15 s: The electrical topology switches from State A
+to State B.
+• t = 20 s: The bus actuation status along with the com-
+munication and electrical topologies revert to State A.
+Bus 9 is connected and Bus 10 is disconnected.
+Furthermore, at each change, half of the buses are ran-
+domly selected and assigned new ZIP load parameters.
+The ZIP load parameters can be found in Appendix B.
+The parameters for the closed-loop controller, as spe-
+cified in Table II, are designed constructively, starting
+from the microgrid subsystems. First, the passivity indices
+for the lines (ρt = 0.01) and loads (ρL = cL = 0.05) are cal-
+culated from Prop. 12 and Prop. 9, respectively. Next, the
+parameters for the power regulator (19) are chosen and the
+DGU passivity indices are calculated from Theorem 10,
+with the optimisation verified for the practically relevant
+intervals v ∈ [200 V, 550 V] and ˆi ∈ [10 A, 350 A]. Note that
+adding the restriction νd,2 ≥ −ρt to the optimisation in
+Theorem 10 ensures that (52) will be met. This yields
+a solution νd,1 = −4.686, νd,2 = −0.01 and ρd = 0.01,
+from which the microgrid supply rate is constructed as
+per Theorem 14. Finally, parameters for the agent PI con-
+trollers are chosen and the weighting function parameters
+are designed using Theorem 17. Note that Theorem 14
+requires strictly passive loads (cL > 0) and Theorem 17
+necessitates leaky integrators (ζc > 0).
+
+12
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+d
+d
+d
+d
+State A
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+d
+d
+d
+d
+State B
+d
+Bus
+Active DGU
+Electrical line
+Communication line
+0
+0.5
+1
+0
+0.5
+1
+Figure 4: Two different states for a 10-bus DC microgrid along with electrical and communication connections. The
+loads at the buses are omitted for clarity.
+0
+5
+10
+15
+20
+25
+340
+360
+380
+400
+420
+Figure 5: Simulated bus voltages with line colours as per
+the legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+0
+40
+80
+Figure 6: Simulated weighted voltage errors and the av-
+erage error of connected agents with agent line colours as
+per the legend in Fig. 4.
+B. Results
+The bus voltages vk shown in Fig. 5 confirm the stability
+of the closed loop results, although the voltages tend to
+be lower than desired, due to the use of leaky integrators.
+The remaining steady-state offset can also be seen in the
+weighted errors plotted in Fig. 6, where the average tends
+towards a non-zero value in each instance (see Remark 6).
+Despite this, the four stage controller reaches a consensus
+on the average of the nonlinear weighted voltage errors.
+Moreover, the advantage of the weighting function can
+be seen at Bus 6 in t ∈ [20 s, 25 s), where a significant
+weighted error only appears in Fig. 6 when the voltage in
+0
+5
+10
+15
+20
+25
+0
+1x104
+2x104
+3x104
+4x104
+Figure 7: Simulated outputs of the local agent controllers
+with line colours as per the legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+0
+10
+20
+30
+40
+Figure 8: Simulated power setpoints with line colours as
+per the legend in Fig. 4.
+Fig. 5 is not close to vRef. Note that the voltages of Buses 9
+and 10 are at 0 V during the respective periods where they
+are disconnected and not actuated.
+In Fig. 7, the outputs of the agent controllers show that
+no synchronisation of the agent controllers are required.
+The agent controller outputs at Buses 1 to 8, which are
+continuously connected to the communication network,
+are near identical. However, the disconnecting buses, e.g.
+Bus 9 after t = 5 s, rapidly diverge from other controllers
+and do not synchronise on reconnect. Despite this, the
+final stage of the controller ensures cooperation of the
+buses, as demonstrated in the power setpoints p∗
+k in Fig. 8.
+When Bus 10 connects at t = 10 s, its setpoint p∗
+k rapidly
+
+13
+0
+5
+10
+15
+20
+25
+340
+360
+380
+400
+420
+Figure 9: Simulated bus voltages with ideal PI controllers
+and with line colours as per the legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+-40
+0
+40
+Figure 10: Simulated weighted voltage errors and the
+average error of connected agents with ideal PI controllers
+and with agent line colours as per the legend in Fig. 4.
+converges to the coordinated common setpoint used by all
+connected agents.
+Although the leaky integrators yield imperfect results
+(see Remark 6 and Fig. 6), this can be mitigated by
+choosing a higher vRef. Indeed, by combining the steady
+state of the agent PI controller (27) with the DDA steady
+state (24), we see that injecting power into the system
+p∗ > 0 results in positive voltage errors. Since we consider
+(strictly) passive loads, increasing vRef is thus a viable
+method for correcting the imperfect results whilst retain-
+ing the advantageous properties of the stability analysis in
+Theorem 17.
+C. Robustness Test
+We
+now
+repeat
+the
+simulation
+described
+in
+Section VII-A with the following changes. 1) Passive
+loads with cL
+= 0 are allowed at all buses, and 2)
+ideal agent PI controllers with ζc = 0 are used. Under
+these conditions, Theorem 17 can no longer be used to
+verify the stability. However, the stability may still be
+verified using classical approaches such as evaluating
+the eigenvalues for the closed loop linearised about the
+equilibrium. Note that the same random seed is used as
+for the results in Section VII-A, allowing for a comparison
+between the scenarios to be made.
+Fig. 9 demonstrates the improved consensus achieved
+by the ideal PI agents, in that the bus voltages are
+closer to vRef at steady state than in Fig. 5. Moreover,
+0
+5
+10
+15
+20
+25
+0
+1x104
+2x104
+3x104
+Figure 11: Simulated outputs of the local agent controllers
+with ideal PI controllers and with line colours as per the
+legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+0
+10
+20
+30
+Figure 12: Simulated power setpoints with ideal PI con-
+trollers and with line colours as per the legend in Fig. 4.
+Fig. 10 shows that perfect consensus is achieved, where
+the average error tends to zero in each case. This figure
+also demonstrates the robustness against communication
+interruptions, as is the case for Bus 10 which, for the
+period t ∈ [5 s, 10 s), is actuated but does not communicate
+with the other buses. Despite this, it is able to accurately
+regulate its own bus voltage (compared to the imperfect
+regulation achieved with leaky integrators as in Fig. 5).
+The lack of leaky integrators is also evident in Fig. 11,
+where the output of the agent controllers stay constant
+when a bus is disconnected and not actuated. Lastly, the
+power setpoints in Fig. 12 converging to a common value
+for the communicating agents confirm the coordination of
+the agents.
+Note that while tests with non-passive loads can also
+yield a stable closed loop, instability can occur when
+the non-passive loads dominate. To address this, a tar-
+geted compensation of non-passive loads is required (see
+Remark 11).
+VIII. Conclusion
+In this paper, we proposed a four-stage distributed
+control structure that achieves power sharing in a DC mi-
+crogrid while ensuring voltage regulation for the voltages
+of both actuated and unactuated buses. We demonstrated
+how the passivity properties of various subsystems can be
+determined and combined these in a stability analysis that
+
+14
+is independent of topological changes, actuation changes,
+bus connections or disconnections and load changes.
+Future work includes the consideration of non-passive
+loads at arbitrary locations in the microgrid and the
+construction of an interface to allow for the presented work
+to be combined with tertiary optimal controllers.
+Appendix A
+Proofs
+Proof of Prop. 8. For the control structure in steady state,
+˙xc = 0 and thus yc is constant. The steady-state output
+(24) of the Stage 4 DDA therefore ensures Objective 2 is
+achieved. Furthermore, consider the steady state of the
+Stage 2 DDA
+ua,s,k = lim
+t→∞ hw(vRef − vk),
+(64)
+lim
+t→∞ ya,2,k = uT
+a,s1N
+N
+= lim
+t→∞
+1
+N
+�
+k∈N
+(vRef − h(vk)) , (65)
+if vk is in equilibrium and where h is obtained by shifting
+hw by vRef. Note that (65) corresponds to the condition of
+(21) in Objective 1. Therefore, ya,2 specifies the regulation
+error of the average weighted voltage error in steady
+state. From the steady state of the agent PI controller
+in (26), we have ζcxc = ya,2. Thus, ideal integrators with
+ζc = 0 ensure that Objective 1 is met exactly. For ζc > 0,
+substitute the PI equilibrium into the output of the agent
+PI controller in (26) to obtain the steady state equation
+xc = 1
+kIc
+�
+yc + kP
+c ya,2
+�
+.
+(66)
+Substitute ζcxc = ya,2 into (24) and simplify to find
+ya,2 =
+ζc
+kIc(1 + ζckPc )yc,
+(67)
+for the steady state. Since the entries of the vector ya,2 and
+thus of xc and yc are the same at steady state. Therefore
+the steady state output for the Stage 4 DDA in (24) gives
+yc = ya,4, which we combine with (67) to obtain the error
+for Objective 1 in (30).
+■
+Proof of Theorem 14. Consider the supply rates which de-
+scribe the actuated and unactuated states, respectively, for
+a given bus k ∈ N
+wM,α,k = (1+νd,1ρd)˜p∗
+α,k˜vα,k − νd,1(˜p∗
+α,k)2 − ρd˜v2
+α,k, (68)
+wM,β,k = −ρL˜v2
+β,k.
+(69)
+These allow the microgrid supply rate in (47) to be
+decomposed according to the actuation states αk
+wM,αβ =
+�
+k∈Nα
+wM,α,k +
+�
+k∈Nβ
+wM,β,k
+=
+�
+k∈N
+(αkwM,α,k + (1 − αk)wM,β,k)
+(70)
+u
+y
+wM,α,k
+wM,α,k
+wM,β,k/ρL
+Figure 13: Comparison of the microgrid supply rate sectors
+in the proof of Theorem 14 if ρd < 0.
+Enlarge the supply rate of the unactuated buses in (69) by
+adding the positive term νL(˜p∗
+β,k)2 for an arbitrarily small
+νL > 0 such that
+wM,β,k ≤ wM,β,k = νL(˜p∗
+β,k)2 − ρL˜v2
+β,k
+≤ wM,β,k
+ρL
+= νL
+ρL
+(˜p∗
+β,k)2 − ˜v2
+β,k
+(71)
+for ρL as in (53). The supply rate wM,β,k/ρL is equivalent
+to the L2 supply rate in Definition 2 and is thus bounded
+by the sector [−
+�
+νL
+ρL ,
+�
+νL
+ρL ] [29, Lemma 4]. Consider now
+the supply rate of the actuated agents (68) narrowed down
+to an IFP sector for the case that ρd < 0, i.e.
+wM,α,k ≥ wM,α,k =
+�
+wM,α,k,
+if ρd ≥ 0,
+˜p∗
+α,k˜vα,k − νd,1(˜p∗
+α,k)2,
+if ρd < 0,
+(72)
+such that wM,α,k is sector bounded by [νd,1, 1
+ρd ] if ρd > 0
+and [νd,1, ∞) if ρd < 0 or if ρd = 0 (see [26, p. 231]). A re-
+lation bewteen wM,α and wM,β/ρL can now be established
+by comparing their respective sector bounds:
+wM,β,k
+ρL
+≤ wM,α,k if
+
+
+
+[−
+�
+νL
+ρL ,
+�
+νL
+ρL ] ⊆ [νd,1, 1
+ρd ], if ρd > 0,
+[−
+�
+νL
+ρL ,
+�
+νL
+ρL ] ⊆ [νd,1, ∞), if ρd ≤ 0,
+(73)
+Since νL can be arbitrarily small, we derive (54) by
+comparing the lower bounds in (73) and note that the
+upper bound relation can be met for any ρd. A visual
+comparison of the sector conditions is made in Fig. 13.
+The combination of (71)–(73) results in
+wM,β,k ≤ wM,β,k ≤ wM,β,k
+ρL
+≤ wM,α,k ≤ wM,α,k.
+(74)
+Therefore, for the microgrid with the storage function SM
+that is dissipative w.r.t. (47), it holds that
+˙SM ≤ wM,αβ ≤
+�
+k∈N
+wM,α,k = wM,
+(75)
+which is found by combining (70) with (74).
+■
+Appendix B
+Simulation Data
+The
+simulation
+parameters
+used
+for
+the
+lines
+in
+Section VII are given in Table III. Furthermore, the
+
+15
+Table III: Rounded Line Lengths
+Line
+Length
+Line
+Length
+Line
+Length
+1 – 2
+1.19 km
+1 – 4
+7.74 km
+2 – 3
+2.23 km
+2 – 4
+7.20 km
+3 – 5
+3.14 km
+3 – 8
+2.82 km
+4 – 5
+3.72 km
+4 – 6
+6.75 km
+4 – 7
+1.16 km
+6 – 7
+4.44 km
+6 – 9
+3.11 km
+7 – 8
+3.69 km
+8 – 10
+1.21 km
+Table IV: Strictly Passive Load Values
+Bus Parameter
+t = 0 s
+t = 5 s
+t = 10 s
+t = 15 s
+t = 20 s
+Z−1 (1/Ω)
+0.103
+0.103
+0.106
+0.106
+0.083
+1
+I (A)
+4.66
+2.15
+-6.08
+-6.08
+14.45
+P (W)
+3599
+-4055
+4133
+4133
+-4927
+Z−1 (1/Ω)
+0.099
+0.099
+0.096
+0.096
+0.080
+2
+I (A)
+-16.09
+-16.09
+19.68
+19.68
+2.49
+P (W)
+3204
+3204
+2659
+2659
+1346
+Z−1 (1/Ω)
+0.128
+0.105
+0.105
+0.105
+0.096
+3
+I (A)
+10.27
+-0.09
+-0.09
+-0.09
+-11.09
+P (W)
+-1479
+-3659
+-3659
+-3659
+3031
+Z−1 (1/Ω)
+0.079
+0.079
+0.079
+0.079
+0.079
+4
+I (A)
+10.15
+10.15
+10.15
+10.15
+10.15
+P (W)
+-2711
+-2711
+-2711
+-2711
+-2711
+Z−1 (1/Ω)
+0.095
+0.095
+0.095
+0.064
+0.107
+5
+I (A)
+-6.64
+-6.64
+-6.64
+16.68
+2.10
+P (W)
+2768
+2768
+2768
+-3798
+4242
+Z−1 (1/Ω)
+0.089
+0.089
+0.106
+0.103
+0.103
+6
+I (A)
+6.87
+6.87
+7.85
+-5.17
+-5.17
+P (W)
+948
+948
+4321
+370
+370
+Z−1 (1/Ω)
+0.065
+0.092
+0.092
+0.118
+0.118
+7
+I (A)
+11.96
+6.51
+6.51
+2.77
+2.77
+P (W)
+-3624
+-3442
+-3442
+-3890
+-3890
+Z−1 (1/Ω)
+0.102
+0.102
+0.086
+0.086
+0.124
+8
+I (A)
+-16.85
+-16.85
+20.71
+20.71
+-4.68
+P (W)
+3529
+3529
+-4773
+-4773
+-3832
+Z−1 (1/Ω)
+0.111
+0.103
+0.109
+0.077
+0.077
+9
+I (A)
+13.79
+-19.74
+9.53
+1.26
+1.26
+P (W)
+-2645
+1830
+4215
+1549
+1549
+Z−1 (1/Ω)
+0.072
+0.100
+0.100
+0.111
+0.111
+10
+I (A)
+7.77
+9.02
+9.02
+10.98
+10.98
+P (W)
+-3538
+-4143
+-4143
+-2795
+-2795
+strictly passive load parameters for the simulation results
+in Section VII-B and the passive load parameters for
+the results in Section VII-C are given in Table IV and
+Table V, respectively. Note that the P parameter for the
+loads in Table V are the same as listed in Table IV.
+References
+[1] B. Lasseter, “Microgrids [distributed power generation],” in
+Proc. 2001 IEEE Power Engineering Society Winter Meeting,
+vol. 1, 2001, pp. 146–149.
+[2] J. J. Justo, F. Mwasilu, J. Lee, and J.-W. Jung, “AC-microgrids
+versus DC-microgrids with distributed energy resources: A re-
+view,” Renewable and Sustainable Energy Reviews, vol. 24, pp.
+387–405, 2013.
+[3] L. Meng, Q. Shafiee, G. F. Trecate, H. Karimi, D. Fulwani,
+X. Lu, and J. M. Guerrero, “Review on control of DC microgrids
+and multiple microgrid clusters,” IEEE J. of Emerging and
+Selected Topics in Power Electron., vol. 5, no. 3, pp. 928–948,
+2017.
+Table V: Passive Load Values, P as in Table IV
+Bus Parameter
+t = 0 s
+t = 5 s
+t = 10 s
+t = 15 s
+t = 20 s
+1
+Z−1 (1/Ω)
+0.091
+0.093
+0.087
+0.087
+0.063
+I (A)
+4.66
+-8.15
+-6.08
+-6.08
+9.71
+2
+Z−1 (1/Ω)
+0.069
+0.069
+0.071
+0.071
+0.046
+I (A)
+-16.09
+-16.09
+19.68
+19.68
+0.20
+3
+Z−1 (1/Ω)
+0.095
+0.082
+0.082
+0.082
+0.059
+I (A)
+8.91
+-7.12
+-7.12
+-7.12
+-11.09
+4
+Z−1 (1/Ω)
+0.038
+0.038
+0.038
+0.038
+0.038
+I (A)
+8.82
+8.82
+8.82
+8.82
+8.82
+5
+Z−1 (1/Ω)
+0.065
+0.065
+0.065
+0.027
+0.078
+I (A)
+-6.64
+-6.64
+-6.64
+15.25
+2.10
+6
+Z−1 (1/Ω)
+0.071
+0.071
+0.089
+0.102
+0.102
+I (A)
+4.04
+4.04
+7.85
+-9.19
+-9.19
+7
+Z−1 (1/Ω)
+0.029
+0.070
+0.070
+0.079
+0.079
+I (A)
+9.04
+0.89
+0.89
+0.58
+0.58
+8
+Z−1 (1/Ω)
+0.075
+0.075
+0.057
+0.057
+0.111
+I (A)
+-16.85
+-16.85
+20.55
+20.55
+-14.31
+9
+Z−1 (1/Ω)
+0.105
+0.102
+0.061
+0.036
+0.036
+I (A)
+10.71
+-19.75
+9.53
+-0.05
+-0.05
+10 Z−1 (1/Ω)
+0.042
+0.091
+0.091
+0.088
+0.088
+I (A)
+2.53
+2.03
+2.03
+8.34
+8.34
+[4] V. Nasirian, S. Moayedi, A. Davoudi, and F. L. Lewis, “Distrib-
+uted cooperative control of DC microgrids,” IEEE Trans. Power
+Electron., vol. 30, no. 4, pp. 2288–2303, 2015.
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+arXiv:2301.04544v1  [cs.GT]  11 Jan 2023
+Optimal Impartial Correspondences
+Javier Cembrano∗
+Felix Fischer†
+Max Klimm‡
+Abstract
+We study mechanisms that select a subset of the vertex set of a directed graph in order to
+maximize the minimum indegree of any selected vertex, subject to an impartiality constraint
+that the selection of a particular vertex is independent of the outgoing edges of that vertex.
+For graphs with maximum outdegree d, we give a mechanism that selects at most d + 1
+vertices and only selects vertices whose indegree is at least the maximum indegree in the
+graph minus one. We then show that this is best possible in the sense that no impartial
+mechanism can only select vertices with maximum degree, even without any restriction
+on the number of selected vertices. We finally obtain the following trade-off between the
+maximum number of vertices selected and the minimum indegree of any selected vertex:
+when selecting at most k vertices out of n, it is possible to only select vertices whose indegree
+is at least the maximum indegree minus ⌊(n − 2)/(k − 1)⌋ + 1.
+1
+Introduction
+Impartial selection is the problem of selecting vertices with large indegree in a directed graph,
+in such a way that the selection of a particular vertex is independent of the outgoing edges of
+that vertex. The problem models a situation where agents nominate one another for selection
+and are willing to offer their true opinion on other agents as long as this does not affect their
+own chance of being selected.
+The selection of a single vertex is governed by strong impossibility results. For graphs with
+maximum outdegree one, corresponding to situations where each agent submits a single nomi-
+nation, every impartial selection rule violates one of two basic axioms [11] and as a consequence
+must fail to provide a non-trivial multiplicative approximation to the maximum indegree. For
+graphs with arbitrary outdegrees, corresponding to situations where each agent can submit mul-
+tiple nominations, impartial rules violate an even weaker axiom and cannot provide a non-trivial
+approximation in a multiplicative or additive sense [1, 8]. These impossibilities largely remain in
+place if rather than a single vertex we want to select any fixed number of vertices, but positive
+results can be obtained if we relax the requirement that the same number of vertices must be
+selected in every graph [4, 18].
+From a practical point of view, the need for such a relaxation should not necessarily be a
+cause for concern. Indeed, situations in the real world to which impartial selection is relevant
+often allow for a certain degree of flexibility in the number of selected agents. The exact number
+of papers accepted to an academic conference is usually not fixed in advance but depends on
+the number and quality of submissions. Best paper awards at conferences are often given in
+overlapping categories, and some awards may only be given if this is warranted by the field of
+candidates. The Fields medal is awarded every four years to two, three, or four mathematicians
+under the age of 40. Examples at the more extreme end of the spectrum of flexibility include
+the award of job titles such as vice president or deputy vice-principal. Such titles can often be
+∗Institut für Mathematik, Technische Universität Berlin, Germany
+†School of Mathematical Sciences, Queen Mary University of London, UK
+‡Institut für Mathematik, Technische Universität Berlin, Germany
+1
+
+given to a large number of individuals at a negligible cost per individual, but should only be
+given to qualified individuals so as not to devalue the title.
+Tamura and Ohseto [18] specifically studied what they call nomination correspondences,
+i.e., rules that may select an arbitrary set of vertices in any graph. For graphs with maximum
+outdegree one a particular such rule, plurality with runners-up, satisfies impartiality and ap-
+propriate versions of the two axioms of Holzman and Moulin [11]. The rule selects any vertex
+with maximum indegree; if there is a unique such vertex, any vertex whose indegree is smaller
+by one and whose outgoing edge goes to the vertex with maximum indegree is selected as well.
+An appropriate measure for the quality of rules that select varying numbers of vertices is the
+difference in the worst case between the best vertex and the worst selected vertex, and we can
+call a rule α-min-additive if the maximum difference, taken over all graphs, between these two
+quantities is at most α. In this terminology, plurality with runners-up is 1-min-additive.
+As Tamura and Ohseto point out, it may be desirable in practice to ensure that the maximum
+number of vertices selected is not too large, a property that plurality with runners-up clearly
+fails. It is therefore interesting to ask whether there exist rules that are α-min-additive and
+never select more than k vertices, for some fixed α and k.
+For graphs with outdegree one,
+Tamura and Ohseto answer this question in the affirmative: a variant of plurality with runners-
+up that breaks ties according to a fixed ordering of the vertices remains 1-min-additive but never
+selects more than two vertices.
+Our Contribution
+Our first result provides a generalization of the result of Tamura and Ohseto
+to graphs with larger outdegrees: for graphs with maximum outdegree d, it is possible to achieve
+1-min-additivity while selecting at most d+1 vertices. For the particular case of graphs with un-
+bounded outdegrees we obtain a slight improvement, by guaranteeing 1-min-additivity without
+ever selecting all vertices. Our second result establishes that 1-min-additivity is best possible,
+thus ruling out the existence of impartial mechanisms that only select vertices with maximum in-
+degree. This holds even when no restrictions are imposed on the number of selected vertices, and
+is shown alongside analogous impossibility results concerning the maximization of the median or
+mean indegree of the selected vertices instead of their minimum indegree. Our third result pro-
+vides a trade-off between the maximum number of vertices selected, where smaller is better, and
+the minimum indegree of any selected vertex, where larger is better: if we are allowed to select
+at most k vertices out of n, we can guarantee α-min-additivity for α = ⌊(n−2)/(k−1)⌋+1. This
+is achieved by removing a subset of the edges from the graph before plurality with runners-up
+is applied, in order to guarantee impartiality while selecting fewer vertices. We do not know
+whether this last result is tight and leave open the interesting question for the optimal trade-off
+between the number and quality of selected vertices.
+Related Work
+Impartiality as a property of an economic mechanism was introduced by
+de Clippel et al. [9], and first applied to the selection of vertices in a directed graph by Alon et al.
+[1] and Holzman and Moulin [11]. Whereas Holzman and Moulin gave axiomatic characteriza-
+tions for mechanisms selecting a single vertex when all outdegrees are equal to one, Alon et al.
+studied the ability of impartial mechanisms to approximate the maximum indegree for any fixed
+number of vertices when there are no limitations on outdegrees.
+Both sets of authors obtained strong impossibility results, which a significant amount of
+follow-up work has since sought to overcome. Randomized mechanisms providing non-trivial
+multiplicative guarantees had already been proposed by Alon et al., and Fischer and Klimm [10]
+subsequently achieved the best possible such guarantee for the selection of one vertex. Starting
+from the observation that worst-case instances for randomized mechanisms have small indegrees,
+Bousquet et al. [5] developed a mechanism that is asymptotically optimal as the maximum
+indegree grows, and Caragiannis et al. [6, 7] initiated the study of mechanisms providing additive
+rather than multiplicative guarantees. Cembrano et al. [8] subsequently identified a deterministic
+2
+
+mechanism that provides non-trivial additive guarantees whenever the maximum outdegree is
+bounded and established that no such guarantees can be obtained with unbounded outdegrees.
+Randomized mechanisms have been also studied from an axiomatic point of view by Mackenzie
+[14, 15].
+Bjelde et al. [4] gave randomized mechanisms with improved multiplicative guarantees for the
+selection of more than one vertex and observed that when selecting at most k vertices rather than
+exactly k, deterministic mechanisms can in fact achieve non-trivial guarantees. An axiomatic
+study of Tamura and Ohseto [18] for the outdegree-one case came to the same conclusion:
+when allowing for the selection of a varying number of vertices, the impossibility result of
+Holzman and Moulin no longer holds.
+Tamura [17] subsequently characterized a mechanism
+proposed by Tamura and Ohseto, which in some cases selects all vertices, as the unique minimal
+mechanism satisfying impartiality, anonymity, symmetry, and monotonicity.
+Impartial mechanisms have finally been proposed for various problems other than selection,
+including peer review [2, 13, 16, 20], rank aggregation [12], progeny maximization [3, 21], and
+network centralities [19].
+2
+Preliminaries
+For n ∈ N, let [n] = {1, 2, . . . , n}, and let
+Gn =
+�
+(V, E) : V = [n], E ⊆ (V × V ) \
+�
+v∈V
+{(v, v)}
+�
+be the set of directed graphs with n vertices and no loops. Let G = �
+n∈N Gn. For G = (V, E) ∈ G
+and v ∈ V , let N +(v, G) = {u ∈ V : (v, u) ∈ E} be the out-neighborhood and N −(v, G) =
+{u ∈ V : (u, v) ∈ E} the in-neighborhood of v in G. Let δ+(v, G) = |N +(v, G)| and δ−(v, G) =
+|N −(v, G)| denote the outdegree and indegree of v in G, and ∆(G) = maxv∈V δ−(v, G) the
+maximum indegree of any vertex in G.
+When the graph is clear from the context, we will
+sometimes drop G from the notation and write N +(v), N −(v), δ+(v), δ−(v), and ∆.
+Let
+top(G) = max{v ∈ V : δ−(v) = ∆(G)} denote the vertex of G with the largest index among
+those with maximum indegree.
+For n ∈ N and d ∈ [n − 1], let Gn(d) = {(V, E) ∈ Gn :
+δ+(v) ≤ d for every v ∈ V } be the set of graphs in Gn with maximum outdegree at most d, and
+G(d) = �
+n∈N Gn(d).
+A k-selection mechanism is then given by a family of functions f : Gn → 2[n], one for
+each n ∈ N, mapping each graph to a subset of its vertices, where we require that |f(G)| ≤ k
+for all G ∈ G. In a slight abuse of notation, we will use f to refer to both the mechanism
+and to individual functions of the family. Given G = (V, E) ∈ G and v ∈ V , let Nv(G) =
+{(V, E′) ∈ G :
+E \ ({v} × V ) = E′ \ ({v} × V )} be the set neighboring graphs of G with
+respect to v, in the sense that they can be obtained from G by changing the outgoing edges
+of v. Mechanism f is impartial on G′ ⊆ G if on this set of graphs the outgoing edges of a
+vertex have no influence on its selection, i.e., if for every graph G = (V, E) ∈ G′, v ∈ V , and
+G′ ∈ Nv(G), it holds that f(G) ∩ {v} = f(G′) ∩ {v}. Given a k-selection mechanism f and
+an aggregator function σ : 2R → R such that σ(∅) = 0 and, for every S ⊆ R with |S| ≥ 1,
+min{x ∈ S} ≤ σ(S) ≤ max{x ∈ S}, we say that f is α-σ-additive on G′ ⊆ G, for α ≥ 0, if for
+every graph in G′ the function σ evaluated on the choice of f differs from the maximum indegree
+by at most α, i.e., if
+sup
+G∈G′
+�
+∆(G) − σ
+�
+{δ−(v, G)}v∈f(G)
+��
+≤ α.
+We will specifically be interested in the cases where σ is the minimum, the median, and the mean,
+and respectively call a mechanism α-min-additive, α-median-additive, and α-mean-additive in
+these cases.
+3
+
+Algorithm 1: Plurality with runners-up
+Input: Digraph G = (V, E) ∈ Gn(1)
+Output: Set S ⊆ V of selected vertices
+Let S = {v ∈ V : δ−(v) = ∆(G)};
+if S = {v} for some v ∈ V then
+S ←− S ∪ {u ∈ V : δ−(u) = ∆(G) − 1 and (u, v) ∈ E}
+end
+Return S
+3
+Plurality with Runners-up
+Focusing on the case with maximum outdegree one, Tamura and Ohseto [18] proposed a mech-
+anism they called plurality with runners-up.
+The mechanism, which we describe formally in
+Algorithm 1, selects all vertices with maximum indegree; if there is a unique such vertex, then
+any vertex with an outgoing edge to that vertex whose indegree is smaller by one is selected as
+well. The idea behind this mechanism is that vertices in the latter category would be among
+those with maximum degree if their outgoing edge was deleted, and thus any impartial mecha-
+nism seeking to select the vertices with maximum degree would also have to select those vertices.
+Plurality with runners-up is impartial on G(1), and in any graph with n vertices selects between
+1 and n vertices whose degree is equal to the maximum degree or the maximum degree minus
+one. It is thus an impartial and 1-min-additive n-selection mechanism on Gn(1) for every n ∈ N.
+It is natural to ask whether a similar additive guarantee can be obtained for more general set-
+tings. In this section, we answer this question in the affirmative, and in particular study for
+which values of n, k, and d there exists an impartial and 1-min-additive k-selection mechanism
+on Gn(d). We will see later, in Section 4, that 1-min-additivity is in fact best possible for all
+cases covered by our result, with the exception of the boundary case where n = 2.
+While Tamura and Ohseto do not limit the maximum number of selected vertices, they
+discuss briefly a modification of their mechanism that retains impartiality and 1-min-additivity
+but selects at most 2 vertices. Instead of all vertices with maximum indegree, the modified
+mechanism breaks ties in favor of a single maximum-degree vertex using a fixed ordering of the
+vertices. In order to guarantee impartiality, the modified mechanism then also selects any vertex
+that would be selected in the graph obtained by deleting the outgoing edge of that vertex. The
+assumption that every vertex has at most one outgoing edge means that at most one additional
+vertex is selected. There thus exists a 1-min-additive k-selection mechanism on G(1) for every
+k ≥ 2.
+Our first result generalizes this mechanism to settings with arbitrary outdegrees, as long
+as the maximum number of selected vertices is large enough. To this end we show that when
+the maximum outdegree is d, to achieve impartiality, at most d vertices have to be selected in
+addition to the one with maximum indegree and highest priority.1 We formally describe the
+resulting mechanism in Algorithm 2, and will refer to it as asymmetric plurality with runners-up
+and denote its output on graph G by P(G). We obtain the following theorem, which generalizes
+the known result for the outdegree-one case.
+Theorem 1. For every n ∈ N, d ∈ [n − 1], and k ∈ {d + 1, . . . , n}, there exists an impartial
+and 1-min-additive k-selection mechanism on Gn(d).
+We will be interested in the following in comparing vertices both according to their indegree
+and to their index, and we will use regular inequality symbols, as well as the operators max and
+1In this mechanism and wherever ties are broken in the rest of the paper, we break ties in favor of greater
+index, so top(G) is the vertex with maximum indegree and highest priority in graph G. Naturally, any other
+deterministic tie-breaking rule could be used instead.
+4
+
+Algorithm 2: Asymmetric plurality with runners-up P(G)
+Input: Digraph G = (V, E) ∈ Gn
+Output: Set S ⊆ V of selected vertices
+Let S = ∅;
+for v ∈ V do
+Let Gv = (V, E \ ({v} × V ));
+if top(Gv) = v then
+S ←− S ∪ {v}
+end
+end
+Return S
+min, to denote the lexicographic order among pairs of the form (δ−(v), v). The following lemma
+characterizes the structure of the set of vertices selected by Algorithm 2, and provides the main
+technical ingredient to the proof of Theorem 1.
+Lemma 1. Let G = (V, E) ∈ G and v ∈ V . Then, v ∈ P(G) if and only if
+(a) for every w ∈ V with (δ−(w), w) > (δ−(v), v) it holds (v, w) ∈ E; and
+(b) one of the following holds:
+(i) δ−(v) = ∆(G); or
+(ii) δ−(v) = ∆(G) − 1 and v > w for every w ∈ V with δ−(w) = ∆(G).
+Proof. We first show that, if v ∈ P(G) for a given graph G, then (a) and (b) follow.
+Let
+G = (V, E) ∈ G, and let v ∈ P(G). To see (a), suppose there is w ∈ V with (δ−(w, G), w) >
+(δ−(v, G), v).
+Since v ∈ P(G), we have v = top(Gv) with Gv = (V, E \ ({v} × V )).
+This
+implies (δ−(v, Gv), v) > (δ−(w, Gv), w) and therefore δ−(w, G) > δ−(w, Gv), because δ−(v, G) =
+δ−(v, Gv). Since G and Gv only differ in the outgoing edges of v, we conclude that (v, w) ∈ E.
+To prove (b), we note that for every w ∈ V we have
+(δ−(v, G), v) = (δ−(v, Gv), v) > (δ−(w, Gv), w) ≥ (δ−(w, G) − 1, w),
+(1)
+where the last inequality comes from the fact that each vertex has at most one incoming edge
+from v. If there is no w ∈ V \ {v} with δ−(w) = ∆(G), the maximum indegree must be that of
+v, so δ−(v) = ∆(G) and (i) follows. Otherwise, for each w ∈ V \ {v} with δ−(w) = ∆(G), (1)
+yields (δ−(v, G), v) > (∆(G) − 1, w). We conclude that either δ−(v, G) > ∆(G) − 1, in which
+case (i) holds, or both δ−(v) = ∆(G) − 1 and v > w, which implies (ii).
+We now prove the other direction.
+Let G = (V, E) ∈ G and v ∈ V such that both (a)
+and (b) hold. Let Gv = (V, E \ ({v} × V )). We have to show that top(Gv) = v, i.e., that
+for every w ∈ V \ {v}, (δ−(v, Gv), v) > (δ−(w, Gv), w).
+Let w be a vertex in V \ {v}.
+If
+(δ−(v, G), v) > (δ−(w, G), w), we can conclude immediately since δ−(v, Gv) = δ−(v, G) and
+δ−(w, Gv) ≤ δ−(w, G). Otherwise, we know from (a) that (v, w) ∈ E and thus δ−(w, Gv) =
+δ−(w, G) − 1. If v satisfies (i), this yields
+δ−(v, Gv) = δ−(v, G) = ∆(G) ≥ δ−(w, G) = δ−(w, Gv) + 1,
+so (δ−(v, Gv), v) > (δ−(w, Gv), w). On the other hand, if v satisfies (ii), then
+δ−(v, Gv) = δ−(v, G) = ∆(G) − 1 ≥ δ−(w, G) − 1 = δ−(w, Gv),
+and v > w implies (δ−(v, Gv), v) > (δ−(w, Gv), w) as well.
+5
+
+4
+2
+1
+6
+5
+3
+∆
+∆ − 1
+Figure 1: Example of a set of vertices selected by Algorithm 2. In this illustration and throughout
+the paper, vertices are arranged vertically according to indegree and horizontally according to
+index, so that vertices on the left are favored in case of ties.
+The vertices selected by the
+mechanism are drawn in white, those not selected in black. Vertices with indegree below ∆ − 1,
+as well as edges incident to such vertices, are not shown. Denoting the graph as G = (V, E),
+and letting Gv = (V, E \ ({v} × V )) for each vertex v, the selected vertices v are those for which
+top(Gv) = v. Specifically, vertices 2, 3, and 6 are not selected because top(G2) = 4, top(G3) = 4,
+and top(G6) = 1.
+Observe that Lemma 1 implies in particular that top(G) ∈ P(G) for every graph G. Figure 1
+provides an example of the characterization given by Lemma 1, in terms of indegrees, tie-
+breaking order, and edges among selected vertices.
+We are now ready to prove Theorem 1.
+Proof of Theorem 1. We show that for every n ∈ N and d ∈ [n − 1], asymmetric plurality with
+runners-up is impartial and 1-min-additive on Gn(d), and that for every G = (V, E) ∈ Gn(d), it
+selects at most d+1 vertices. If this is the case, then for every k ∈ {d+1, . . . , n} the mechanism
+would satisfy the statement of the theorem. Therefore, let n and d be as mentioned.
+Impartiality follows from the definition of the mechanism, because the outgoing edges of a
+vertex are not taken into account when deciding whether the vertex is taking part on the selected
+set or not. If we let G = (V, E), v ∈ V , and G′ = (V, E′) ∈ Nv(G), then the graphs Gv and
+G′
+v constructed when running the mechanism with each of these graphs G and G′ as an input,
+respectively, are the same because by definition of Nv(G) we have E \({v}×V ) = E′ \({v}×V ).
+Since v ∈ P(G) ⇔ top(Gv) = v, and v ∈ P(G′) ⇔ top(G′
+v) = v, we conclude v ∈ P(G) ⇔ v ∈
+P(G′).
+To see that the mechanism is 1-min-additive, let G ∈ Gn(d) and first note that P(G) ̸= ∅
+since Lemma 1 implies that top(G) ∈ P(G). From this lemma we also know that for every
+v ∈ P(G), δ−(v) ≥ ∆(G) − 1. We conclude that min{{δ−(v)}v∈P(G)} ≥ ∆(G) − 1, and since
+this holds for every G ∈ Gn(d), the mechanism is 1-min-additive.
+Finally, let G = (V, E) ∈ Gn(d), and suppose that |P(G)| > d + 1.
+If we denote vL =
+argminv∈P(G){(δ−(v), v)}, from Lemma 1 we know that (vL, w) ∈ E for every w ∈ V with
+(δ−(w), w) > (δ−(vL), vL), thus δ+(vL) ≥ |P(G)| − 1 > d, a contradiction. We conclude that
+|P(G)| ≤ d + 1.
+The following result, concerning mechanisms that may select an arbitrary number of vertices,
+follows immediately from Theorem 1.
+Corollary 1. For every n ∈ N, there exists an impartial and 1-min-additive n-selection mecha-
+nism on Gn.
+On Gn, i.e., in the case of unbounded outdegrees, this result can in fact be improved slightly
+to guarantee 1-min-additivity while selecting only at most n − 1 vertices. The improvement
+is achieved by a more intricate version of asymmetric plurality with runners-up, which we call
+asymmetric plurality with runners-up and pivotal vertices. We formally describe this mechanism
+6
+
+Algorithm 3: Asymmetric plurality with runners-up and pivotal vertices PP(G)
+Input: Digraph G = (V, E) ∈ Gn
+Output: Set S ⊆ V of selected vertices with |S| ≤ n − 1
+Let S ←− ∅;
+for u ∈ P(G) do
+if for every v ∈ P(G) \ {u} there exists Guv ∈ Nu(G) such that v /∈ P(Guv) then
+S ←− S ∪ {u}
+end
+end
+Return S
+in Algorithm 3 and denote its output for graph G by PP (G).
+Given a graph G = (V, E),
+call a vertex u ∈ P(G) pivotal for v ∈ P(G) if there exists a graph Guv ∈ Nu(G) such that
+v /∈ P(Guv), i.e., if the outgoing edges of u can be changed in such a way that v is no longer
+selected by asymmetric plurality with runners-up. Asymmetric plurality with runners-up and
+pivotal vertices then selects every vertex in P(G) that is pivotal for every other vertex in P(G).
+The mechanism turns out to inherit impartiality and 1-min-additivity, and to never select all
+vertices.
+Theorem 2. For every n ∈ N and k ∈ {n − 1, n}, there exists an impartial and 1-min-additive
+k-selection mechanism on Gn.
+Proof. We show that for every n ∈ N, asymmetric plurality with runners-up and pivotal vertices
+is impartial and 1-min-additive on Gn and that for every G = (V, E) ∈ Gn, it selects at most
+n − 1 vertices. Let n ∈ N be an arbitrary value.
+To see that the mechanism is impartial, let G = (V, E) ∈ Gn, u ∈ PP (G), and G′ = (V, E′) ∈
+Nu(G). We show in the following that u ∈ PP (G′), and since the graphs G and G′ are chosen
+arbitrarily, their roles can be inverted and this is enough to conclude that the mechanism is
+impartial. We first note that u ∈ P(G) because PP (G) ⊆ P(G), thus impartiality of asymmetric
+plurality with runners-up proven in Theorem 1 implies u ∈ P(G′). If P(G′) = {u}, then the
+condition in the mechanism holds trivially for this vertex, so u ∈ PP (G′) and we conclude.
+Otherwise, let v ∈ P(G′) \ {u} be an arbitrary vertex selected by asymmetric plurality with
+runners-up other than u. Since u ∈ PP (G), we have that either (a) v /∈ P(G), or (b) v ∈ P(G)
+and there exists Guv = (V, Euv) ∈ Nu(G) such that v /∈ P(Guv). If (a) holds, taking G′
+uv = G,
+which belongs to Nu(G′) because of the assumption that G′ ∈ Nu(G), we have that v /∈ P(G′
+uv).
+If (b) holds, taking G′
+uv = Guv, which belongs to Nu(G′) since Nu(G′) = Nu(G), we have that
+v /∈ P(G′
+uv). In either case, we conclude that there exists G′
+uv ∈ Nu(G′) such that v /∈ P(G′
+uv).
+Since this argument is valid for every v ∈ P(G′) \ {u}, we conclude that u ∈ PP (G′).
+To see that the mechanism is 1-min-additive, it is enough to show that it always selects a
+vertex, since for every G ∈ G it selects a subset of P(G) and from Theorem 1 we know that this
+set contains vertices with indegrees in {∆(G), ∆(G)−1}. To this purpose we let G = (V, E) ∈ Gn
+and introduce some additional notation. Let Si = {v ∈ P(G) : δ−(v) = ∆(G) − i} and ni = |Si|
+for i ∈ {0, 1}, and denote
+vH = argmaxv∈P(G){(δ−(v, G), v)} = top(G),
+vL = argminv∈P(G){(δ−(v, G), v)}.
+From Lemma 1, we know that P(G) = S0 ∪ S1, that (vL, v) ∈ E for every v ∈ P(G) \ {vL},
+and that u > v for each u ∈ S1, v ∈ S0. We now distinguish two cases according to the edges
+between vertices in P(G).
+If (vH, v) ∈ E for every v ∈ P(G) \ {vH}, then we claim that defining G′ = (V, E \ ({vH} ×
+V )) ∈ NvH(G) it holds v /∈ P(G′) for every v ∈ P(G) \ {vH}. If this is true, it is clear that
+7
+
+∆
+∆ − 1
+∆ − 2
+G = (V, E)
+G′ = (V, E \ ({2} × V ))
+2
+1
+4
+3
+2
+1
+4
+3
+Figure 2: Illustration of the fact that the set of vertices selected by Algorithm 3 is non-empty if
+(vH, v) ∈ E for every v ∈ P(G) \ {vH}. Vertices selected by asymmetric plurality with runners-
+up are drawn in white. Denoting the graph on the left as G = (V, E), where vH = 2, and defining
+G′ = (V, E \ ({2} × V )) ∈ N2(G), we have that {1, 3, 4} ∩ P(G′) = ∅, and thus 2 ∈ PP (G).
+vH ∈ PP (G) and thus PP (G) ̸= ∅. We now prove the claim. First, note that vH ∈ P(G′)
+since vH = top(G′) and Lemma 1 ensures top(G′) ∈ P(G′). This comes from the fact that
+δ−(vH, G′) = δ−(vH, G) and δ−(v, G′) ≤ δ−(v, G) for every v ∈ V \ {vH}, together with vH =
+top(G). Moreover, for every v ∈ S0 \ {vH} it holds δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 1 =
+∆(G′) − 1 and v < vH, so condition (b) in Lemma 1 does not hold for v and thus v /∈ P(G′).
+Analogously, for every v ∈ S1 it holds δ−(v, G′) = δ−(v, G)−1 = δ−(vH, G′)−2 = ∆(G′)−2, so
+condition (b) in Lemma 1 does not hold for v either, and thus v /∈ P(G′). This allows to conclude
+the claim and the fact that PP (G) is non-empty for this case. This argument is illustrated in
+Figure 2.
+Now we consider the case where there is a vertex ¯v ∈ P(G) such that (vH, ¯v) /∈ E, and
+we claim that defining G′ = (V, (E \ ({vL × V })) ∪ (vL, vH)) ∈ NvL(G) it holds v /∈ P(G′)
+for every v ∈ P(G) \ {vL, vH}, whereas defining G′′ = (V, E \ (vL, vH)) ∈ NvL(G) it holds
+vH /∈ P(G′′). If this is true, then vL ∈ PP (G) and PP (G) ̸= ∅. We now prove the claim.
+First, note that vH ∈ P(G′) for the same reason as before, since δ−(vH, G′) = δ−(vH, G) and
+δ−(v, G′) ≤ δ−(v, G) for every v ∈ V \ {vH}.
+Moreover, for every v ∈ S0 \ {vH} it holds
+δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 1 = ∆(G′) − 1 and v < vH, so condition (b) in
+Lemma 1 does not hold for v and thus v /∈ P(G′). Analogously, for every v ∈ S1 \ {vL} it holds
+δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 2 = ∆(G′) − 2 so condition (b) in Lemma 1 does not
+hold for v and thus v /∈ P(G′). This allows to conclude the claim for G′. In the case of G′′, we
+can write the following chain of inequalities,
+(δ−(¯v, G′′), ¯v) = (δ−(¯v, G), ¯v) > (δ−(vH, G) − 1, vH) = (δ−(vH, G′′), vH),
+where the equalities hold because of the definition of G′′ and the inequality by condition (b)
+in Lemma 1, given that ¯v ∈ P(G).
+Since (vH, ¯v) /∈ E, we conclude from condition (a) in
+Lemma 1 that vH /∈ P(G′′), and therefore the claim for G′′ follows. This argument is illustrated
+in Figure 3.
+Finally, we show that the mechanism selects at most n − 1 vertices. Let G = (V, E) ∈ Gn.
+Since PP (G) ⊆ P(G), if |P(G)| ≤ n − 1 this is immediate. We thus suppose in what follows
+that |P(G)| = n. In particular, Lemma 1 implies (v, vH) ∈ E for every v ∈ V \ {vH}, thus
+∆(G) = n − 1, and δ−(v) ≥ n − 2 for every v ∈ V . If S1 = ∅, then δ−(v) = n − 1 for every
+v ∈ V , i.e., G is the complete graph. In this case, vH = n and we claim that v /∈ PP (G) for
+each v ∈ V \ {n}, thus |PP (G)| ≤ 1. This comes from the fact that, for every v ∈ V \ {n}
+and every G′ = (V, E′) ∈ Nv(G) it holds n ∈ P(G′). To see this, note that (n, v) ∈ E′ for
+every v ∈ V \ {n}, δ−(n, G′) ≥ n − 2 = ∆(G′) − 1, and n > v for every v ∈ V \ {n}, so
+Lemma 1 ensures n ∈ P(G′). If S1 ̸= ∅, then there is at least one vertex with outdegree less
+8
+
+∆
+∆ − 1
+∆ − 2
+G′ = (V, E \ {(3, 1), (3, 4)})
+G′′ = (V, E \ {(3, 2)})
+∆
+∆ − 1
+G = (V, E)
+2
+1
+4
+3
+2
+1
+4
+3
+2
+1
+4
+3
+Figure 3: Illustration of the fact that the set of vertices selected by Algorithm 3 is non-empty if
+(vH, ¯v) /∈ E for some ¯v ∈ P(G) \ {vH}. Vertices selected by asymmetric plurality with runners-
+up are drawn in white. Denoting the graph at the top by G = (V, E), where vH = 2, vL = 3,
+and ¯v = 4, and defining G′ = (V, E \ {(3, 1), (3, 4)}) ∈ N3(G), we have that {1, 4} ∩ P(G′) = ∅,
+whereas defining G′′ = (V, (E \ {(3, 2)}) ∈ N3(G) we have that 2 /∈ P(G′′). We conclude that
+3 ∈ PP (G).
+or equal than n − 2. Let u be an arbitrary vertex with δ+(u) ≤ n − 2, and let ¯v ∈ S1 be the
+vertex with highest index such that (u, ¯v) /∈ E, i.e., ¯v = max{V \ N +(u)}. Since u ∈ PP (G),
+there exists G′ = (V, E′) ∈ Nu(G) such that ¯v /∈ P(G′). From Lemma 1, this implies that
+there exists ¯w ∈ V such that either (a) (δ−( ¯w, G′), w) > (δ−(¯v, G′), ¯v) and (¯v, ¯w) /∈ E′, or
+(b) δ−( ¯w, G′) > δ−(¯v, G′) and ¯w > ¯v.
+Since ¯v ∈ P(G), we know from this same lemma
+that if (a) holds, (δ−( ¯w, G), w) < (δ−(¯v, G), ¯v) because of having ¯w /∈ N +(¯v, G) = N +(¯v, G′);
+and similarly, if (b) holds, δ−( ¯w, G) ≤ δ−(¯v, G) because of having ¯w > ¯v.
+In either case,
+since δ−(¯v, G) ≤ δ−(¯v, G′), we conclude that δ−( ¯w, G′) > δ−( ¯w, G), and therefore (u, ¯w) /∈ E.
+If (a) holds, this is a contradiction because we would have {u, ¯v} ∩ N −( ¯w, G) = ∅ and thus
+δ−( ¯w, G) ≤ n − 3.
+If (b) holds, we reach a contradiction as well, because we would have
+¯w ∈ V \ N +(u, G) and ¯w > ¯v, but we chose ¯v to be the maximum of this set.
+4
+An Impossibility Result
+When we established the existence of an impartial and 1-min-additive k-selection mechanism on
+G(d) whenever k ≥ d+1, we claimed this result to be best possible in the sense that the additive
+guarantee cannot be improved. We will prove this claim, that impartiality is incompatible with
+the requirement to only select vertices with maximum indegree, as a corollary of a more general
+result.
+While selecting only vertices with maximum indegree is a natural goal for mechanisms that
+select varying numbers of vertices, other natural objectives exist for such mechanisms such as
+maximizing the median or mean indegree of the selected vertices. For both of these objectives,
+the mechanisms discussed in the previous section immediately provide upper bounds: if a k-
+selection mechanism always selects one vertex with maximum indegree and is α-min-additive
+9
+
+then it is clearly α-median-additive and
+� k−1
+k α
+�
+-mean-additive; Theorem 1 thus implies the ex-
+istence of a 1-median-additive and k−1
+k -mean-additive k-selection mechanism on G(d), whenever
+k ≥ d + 1. To improve on 1-median-additivity, it would be acceptable to select vertices with low
+indegree as long as a greater number of vertices with maximum indegree is selected at the same
+time. To improve on k−1
+k -mean-additivity, it would suffice to select more than one vertex with
+maximum indegree whenever this is possible, and to otherwise select only a sublinear number
+in k of vertices with indegree equal to the maximum indegree minus one. The following result
+shows that no such improvements are possible.
+Theorem 3. Let n ∈ N, n ≥ 3, k ∈ [n], and d ∈ [n − 1]. Let f be an impartial k-selection
+mechanism. If f is α1-median-additive on Gn(d), then α1 ≥ 1/2(1+1(d ≥ 3)). If f is α2-mean-
+additive on Gn(d), then α2 ≥
+� d+1
+2
+�
+/
+�� d+1
+2
+�
++ 1
+�
+.
+Proof. Let n, k, and d be as in the statement of the theorem. In the following we suppose that
+there is an impartial k-selection mechanism f which is either α1-median-additive on Gn(d) with
+α1 < 1/2(1 + 1(d ≥ 3)), or α2-mean-additive on Gn(d) with α2 <
+� d+1
+2
+�
+/
+�� d+1
+2
+�
++ 1
+�
+.
+We first prove the result for the case d = 1. We consider the graph G = (V, E) ∈ Gn(1)
+with E = {(1, 2), (2, 3), (3, 1)}, consisting of a 3-cycle and n − 3 isolated vertices. We consider
+as well, for v ∈ {1, 2, 3}, the graph Gv = (V, Ev) where v deviates from the 3-cycle by changing
+its outgoing edge to the previous vertex in the cycle, i.e.,
+E1 = {(1, 3), (2, 3), (3, 1)}, E2 = {(1, 2), (2, 1), (3, 1)}, E3 = {(1, 2), (2, 3), (3, 2)}.
+Since f is α1-median-additive with α1 < 1/2 or α2-mean-additive with α2 < 1/2, we have that
+f(G1) = {3}, f(G2) = {1}, and f(G3) = {2}. In particular, for v ∈ {1, 2, 3}, v /∈ f(Gv). Since
+for each v ∈ {1, 2, 3} it holds Ev \ ({v} × V ) = E \ ({v} × V ), we conclude by impartiality
+that v /∈ f(G), and thus f(G) ∩ {1, 2, 3} = ∅.
+This implies that both the median and the
+mean indegree of the vertices in f(G) are 0, which contradicts the additive guarantee of this
+mechanism because ∆(G) = 1.
+In the following, we assume d ≥ 2. We denote D = [d + 1] and consider in what follows two
+families of graphs with n vertices, Kv for each v ∈ D and Kuv for each u, v ∈ D, u ̸= v. They
+are constructed from a complete subgraph on D but deleting the outgoing edges of v, in the
+case of Kv, and the outgoing edges of u and v, in the case of Kuv. All the other vertices remain
+isolated. Formally, taking V = [n] we define
+Kv = (V, (D \ {v}) × D) for every v ∈ D,
+Kuv = (V, (D \ {u, v}) × D) for every u, v ∈ D with u ̸= v.
+If there is v ∈ D such that v /∈ f(Kv), then
+median
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤ d − 1 = ∆(Kv) − 1,
+mean
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤ d − 1 = ∆(Kv) − 1,
+which is a contradiction, so the result follows immediately. Therefore, in the following we assume
+that for every v ∈ D we have v ∈ f(Kv). We claim that for every v ∈ D,
+|{u ∈ D \ {v} : u ∈ f(Kv)}| ≥
+�d + 1
+2
+�
+.
+Let us see why the result follows if the claim holds. If this is the case, f selects one vertex with
+maximum indegree d in Kv and at least
+� d+1
+2
+�
+vertices with indegree d − 1. This yields both
+median
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤
+� d − 1
+2
+if d = 2
+d − 1
+otherwise,
+10
+
+and
+mean
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤ d + (d − 1)
+� d+1
+2
+�
+� d+1
+2
+�
++ 1
+= d −
+� d+1
+2
+�
+� d+1
+2
+�
++ 1,
+which is a contradiction since ∆(Kv) = d.
+Now we prove the claim. Suppose that for every v ∈ D we have v ∈ f(Kv) and
+|{u ∈ D \ {v} : u ∈ f(Kv)}| <
+�d + 1
+2
+�
+.
+(2)
+Let v ∈ D and u ∈ D \ {v} such that u /∈ f(Kv). Observing that
+((D \ {v}) × D) \ ({u} × V ) = ((D \ {u, v}) × D) \ ({u} × V ),
+we obtain from impartiality that u /∈ f(Kuv). From the bounds on α1 or α2 that f satisfies
+by assumption, this mechanism has to select a vertex with maximum indegree in this graph;
+otherwise, both the median and the mean of the selected set would be at most ∆(Kuv) − 1.
+Since δ−(w) < ∆(Kuv) for every w /∈ {u, v}, it holds v ∈ f(Kuv). Using impartiality once again,
+we conclude v ∈ f(Ku). We have shown the following property:
+For every u, v ∈ D : u /∈ f(Kv) =⇒ v ∈ f(Ku).
+(3)
+Consider now the graph H = (D, F), where for each u, v ∈ D with u ̸= v, (u, v) ∈ F if and
+only if u /∈ f(Kv). Property (2) implies that
+δ−(v, H) > d −
+�d + 1
+2
+�
+⇐⇒ δ−(v, H) ≥ d + 1 −
+�d + 1
+2
+�
+for each v ∈ D. In particular, there has to be a vertex v∗ ∈ D such that δ+(v∗, H) ≥ d + 1 −
+⌊(d + 1)/(2)⌋ as well. For this vertex we have
+δ+(v∗, H) + δ−(v∗, H) ≥ 2
+�
+d + 1 −
+�d + 1
+2
+��
+≥ d + 1.
+Since H has d+1 vertices, this implies the existence of w∗ ∈ D for which {(v∗, w∗), (w∗, v∗)} ⊂ F,
+i.e., both v∗ /∈ f(Kw∗) and w∗ /∈ f(Kv∗). This contradicts (3), so we conclude the proof of the
+claim and the proof of the theorem.
+Figure 4 provides an illustration of Theorem 3 for the case where n = 3, Figure 5 for the
+case where n = 4.
+The median of any set of numbers is an upper bound on their minimum. Therefore, if no
+impartial mechanism exists that is α-median-additive on G′ ⊆ G for α < ¯α, then no impartial
+mechanism can exist that is α-min-additive on G′ for α < ⌈¯α⌉. We thus obtain the following
+impossibility result, which we have claimed previously.
+Corollary 2. Let n ∈ N, n ≥ 3, and k ∈ [n]. Let f be an α-min-additive impartial k-selection
+mechanism on Gn. Then α ≥ 1.
+The impossibility results imply that for k ≥ d + 1, the mechanisms of Section 3 are best
+possible for the minimum and median objectives except in a few boundary cases. When n = 2,
+selecting each of the two vertices if and only if it has an incoming edge is impartial and achieves
+0-min-additivity and 0-median-additivity. When n = 3, it is possible to select in an impartial
+way at least one vertex with maximum indegree and at most one vertex with indegree equal
+to the maximum indegree minus one, thus guaranteeing 1/2-median-additivity. For the mean
+objective, the mechanisms of Section 3 are best possible asymptotically under the additional
+assumption that k = O(d).
+11
+
+1
+2
+3
+1
+1
+2
+3
+2
+1
+2
+3
+3
+1
+2
+3
+Figure 4: Counterexample to the existence of an impartial 3-selection mechanism that is α-
+median-additive or α-mean-additive on G3 for α < 1/2. Vertices drawn in white have to be
+selected, vertices in black cannot be selected. For the graphs at the top, on the left, and on
+the right, this follows from α-median-additivity or α-mean-additivity for α < 1/2. An arrow
+with label v from one graph to another indicates that one can be obtained from the other by
+changing the outgoing edges of vertex v; by impartiality, the vertex thus has to be selected in
+both graphs or not selected in both graphs. It follows that no vertices are selected in the graph
+at the center, a contradiction to the claimed additive guarantee.
+It is worth pointing out that the proof of the impossibility result uses graphs in which some
+vertices, in particular those with maximum indegree, do not have any outgoing edges. However,
+the impossibility extends naturally to the case where this cannot happen, corresponding to the
+practically relevant case in which abstentions are not allowed, as long as n ≥ 4 and d ≥ 3. For
+this it is enough to define D = [d], add a new vertex with outgoing edges to every vertex in D
+and incoming edges from the vertices in D which do not have any outgoing edge, and construct
+a cycle containing the vertices in V \ D.
+5
+Trading Off Quantity and Quality
+We have so far given impartial selection mechanisms for settings where the maximum outdegree d
+is smaller than the maximum number k of vertices that can be selected, and have shown that
+the mechanisms provide best possible additive guarantees in such settings. We will now consider
+settings where d ≥ k, such that asymmetric plurality with runners-up selects too many vertices
+and therefore cannot be used directly. We obtain the following result.
+Theorem 4. For every n ∈ N and k ∈ {2, . . . , n}, there exists an impartial and (⌊(n − 2)/(k −
+1)⌋ + 1)-min-additive k-selection mechanism on Gn.
+The result is obtained by a variant of asymmetric plurality with runners-up in which some
+edges are deleted before the mechanism is run. In principle, deleting a certain number of edges
+can affect the additive guarantee by the same amount, if all of the deleted edges happen to
+be directed at the same vertex. By studying the structure of the set of vertices selected by the
+mechanism, we will instead be able to delete edges to distinct vertices and thus keep the negative
+impact on the additive guarantee under control.
+The modified mechanism, which we call asymmetric plurality with runners-up and edge
+deletion, is formally described in Algorithm 4.
+It deletes any edges from a vertex to the
+⌊(n−2)/(k−1)⌋ vertices preceding that vertex in the tie-breaking order, and applies asymmetric
+12
+
+1
+2
+3
+4
+2
+3
+1
+2
+3
+4
+4
+1
+2
+3
+4
+4
+1
+2
+3
+4
+1
+1
+2
+3
+4
+1
+1
+2
+3
+4
+1
+2
+3
+4
+2
+3
+1
+2
+3
+4
+Figure 5: Counterexample to the existence of an impartial 4-selection mechanism that is α1-
+median-additive on G4(3) for α1 < 1 or α2-mean-additive on G4 for α2 < 2/3. Vertices drawn
+in white have to be selected, vertices in black cannot be selected, and vertices in gray may
+or may not be selected. For the graph on the left, this follows from α1-median-additivity for
+α1 < 1 or α2-mean-additivity for α2 < 2/3: under these assumptions at most one of the vertices
+with indegree 2 can be selected, which without loss of generality we can assume to be vertex 1.
+For the other graphs, it then follows by impartiality, and for the graph on the right yields a
+contradiction to the claimed additive guarantees.
+plurality with runners-up to the resulting graph. The following lemma shows that without such
+edges, the maximum number of vertices selected is reduced to k.
+Lemma 2. Let n ∈ N, k ∈ {2, . . . , n}, and r ∈ N with r ≥ ⌊(n−2)/(k−1)⌋. Let G = (V, E) ∈ Gn
+be such that for every u ∈ {1, . . . , n − 1} and every v ∈ {u + 1, . . . , min{u + r, n}}, (u, v) /∈ E.
+Then, |P(G)| ≤ k.
+Proof. As in the proof of Theorem 2, we let Si = {v ∈ P(G) : δ−(v) = ∆(G) − i} and ni = |Si|
+for i ∈ {0, 1}, and now we denote its elements in increasing order by vi
+j for j ∈ [ni], i.e.,
+Si = {vi
+j}ni
+j=1 with vi
+1 < vi
+2 · · · < vi
+ni for each i ∈ {0, 1}.
+From Lemma 1, we know that P(G) = S0 ∪S1, that for i ∈ {0, 1} we have (vi
+j, vi
+k) ∈ E for every
+j, k with j < k, and that v1
+1 > v0
+n0. This allows to define, for i ∈ {0, 1},
+¯Si = {v ∈ V \ Si : vi
+1 < v < vi
+ni},
+¯ni = | ¯Si|,
+such that ¯S0 ∩ ¯S1 = ∅.
+Fix i ∈ {0, 1} and suppose that ni ≥ 2. Combining both the fact that (vi
+j, vi
+k) ∈ E for every
+j, k with j < k, and that for every u ∈ {1, . . . , n−1} and v ∈ {u+1, . . . , min{u+r, n}}, (u, v) /∈
+13
+
+Algorithm 4: Asymmetric plurality with runners-up and edge deletion PD(G)
+Input: Digraph G = (V, E) ∈ Gn, k ∈ {2, . . . , n}
+Output: Set S ⊆ V of selected vertices with |S| ≤ k
+Let r = ⌊(n − 2)/(k − 1)⌋ ;
+// number of outgoing edges to remove
+Let R = �n−1
+u=1
+�min{u+r,n}
+v=u+1
+{(u, v)} ;
+// edges to be removed
+Let ¯G = (V, E \ R);
+Return P( ¯G)
+v0
+n0
+. . .
+v0
+2
+v0
+1
+v1
+n1
+. . .
+v1
+2
+v1
+1
+∆
+∆ − 1
+. . .
+� �� �
+≥r
+. . .
+. . .
+� �� �
+≥r
+. . .
+� �� �
+≥r
+. . .
+� �� �
+≥r
+. . .
+. . .
+� �� �
+≥r
+. . .
+� �� �
+≥r
+S1
+¯S1
+S0
+¯S0
+Figure 6: Illustration of Lemma 2. There are no edges from a vertex to any of the r vertices
+to its left, which means that for each vertex in S0 or S1, except for the left-most vertex, there
+exist are at least r vertices outside these sets. Such vertices are not arranged according to their
+indegrees, and edges from vertices in S1 to every vertex in S0 have been omitted for clarity.
+E, we have that for every j ∈ [ni − 1] it holds vi
+j+1 − vi
+j ≥ r + 1.
+Summing over j yields
+vi
+ni − vi
+1 ≥ (ni − 1)(r + 1), hence
+¯ni = vi
+ni − vi
+1 + 1 − ni ≥ (ni − 1)(r + 1) + 1 − ni = (ni − 1)r,
+where the first equality comes from the definition of the set ¯Si. This implies ni ≤ 1 + ¯ni/r. We
+can now lift the assumption ni ≥ 2, since when ni = 1 we have ¯ni = 0 and the inequality holds
+as well, and write the following chain of inequalities:
+|P(G)| = n0 + n1 ≤ 2 + ¯n0 + ¯n1
+r
+≤ 2 + n − |P(G)|
+r
+,
+where the last inequality comes from the fact that all the sets S0, S1, ¯S0, ¯S1 are disjoint and
+therefore their cardinalities sum up to at most n. This bounds the number of selected vertices
+as |P(G)| ≤ (2r + n)/(r + 1).
+Suppose now that |P(G)| ≥ k + 1. Using the previous bound, this yields
+2r + n ≥ (k + 1)(r + 1) ⇐⇒ r ≤ n − k − 1
+k − 1
+= n − 2
+k − 1 − 1,
+which contradicts the lower bound on r in the statement of the lemma.
+Figure 6 illustrates the argument and notation of Lemma 2. We are now ready to prove
+Theorem 4.
+Proof of Theorem 4. We show that Algorithm 4 satisfies the conditions of the theorem.
+Let
+n ∈ N and k ∈ {2, . . . , n}. Impartiality follows from the fact that Algorithm 2 is impartial,
+thus the potential deletion of outgoing edges of a given vertex cannot affect the fact of selecting
+14
+
+this vertex or not. Formally, if G = (V, E), v ∈ V and G′ = (V, E′) ∈ Nv(G), then defining
+¯G = (V, ¯E) and ¯G′ = (V, ¯E′) as the graphs constructed when running Algorithm 4 with G and
+G′ as input graphs, respectively, we have
+¯E \ ({v} × V ) = (E \ ({v} × V )) \
+
+
+n−1
+�
+u=1
+min{u+r,n}
+�
+w=u+1
+{(u, w)}
+
+
+= (E′ \ ({v} × V )) \
+
+
+n−1
+�
+u=1
+min{u+r,n}
+�
+w=u+1
+{(u, w)}
+
+
+= ¯E′ \ ({v} × V ),
+where we use that G′ ∈ Nv(G). Impartiality then follows directly from impartiality of plurality
+with runners-up. For the following, let G = (V, E) ∈ Gn and define r and ¯G as in the mechanism.
+Since the first step of the mechanism ensures that for every u ∈ {1, . . . , n − 1} and every
+v ∈ {u + 1, . . . , min{u + r, n}}, (u, v) /∈ E, Lemma 2 implies that |PD(G)| = |P( ¯G)| ≤ k.
+Finally, in order to show the additive guarantee we first note that, for every v ∈ V, δ−(v, G) ≤
+δ−(v, ¯G) + r, since at most |{v − r, . . . , v − 1} ∩ V | ≤ r incoming edges of v are deleted when
+defining ¯G from G. In particular, ∆(G) ≤ ∆( ¯G) + r. Using this observation and denoting v∗ ∈
+argminv∈PD(G){δ−(v, G)} an arbitrary element with minimum indegree among those selected by
+asymmetric plurality with runners-up and edge deletion, we obtain that
+δ−(v∗, G) ≥ δ−(v∗, ¯G) ≥ ∆( ¯G) − 1 ≥ ∆(G) − r − 1,
+where the second inequality comes from Lemma 1, since v∗ belongs to P( ¯G). We conclude that
+the mechanism is (r + 1)-min-additive for r = ⌊(n − 2)/(k − 1)⌋.
+It is easy to see that the previous analysis is tight from a graph G = (V, E) where exactly
+r = ⌊(n − 2)/(k − 1)⌋ incoming edges of the top-voted vertex are deleted, and a vertex with the
+second highest indegree u such that u > top(G), (u, top(G)) ∈ E, and δ−(u) = ∆(G) − r − 1 is
+selected. However, we do not know whether the tradeoff provided by Theorem 4 is best possible
+for any impartial mechanism, and the question for the optimum tradeoff is an interesting one.
+Currently, when d ≥ k a gap remains between the upper bound of ⌊(n − 2)/(k − 1)⌋ + 1 and a
+lower bound of 1, which is relatively large when the number k of vertices that can be selected
+is small. We may, alternatively, also ask for the number of vertices that have to be selected in
+order to guarantee 1-min-additivity. Currently, the best upper bound on this number is n − 1.
+In addition to the question about the performance of the mechanism introduced in this
+section, the sole fact that sometimes it does not select vertices with indegree strictly higher than
+the one of other selected vertices may seem unfair. Unfortunately, this is unavoidable whenever
+d ≥ k and α-min-additivity is imposed for some α < d, as one can see from a graph consisting
+of a complete subgraph on d + 1 vertices and n − (d + 1) isolated vertices. For any k-selection
+mechanism, a vertex in the complete subgraph is not selected, and impartiality forces us to not
+select it either when its outgoing edges are deleted and it is the unique top-voted vertex.
+Acknowledgments
+The authors have benefitted from discussions with David Hannon. Re-
+search was supported by the Deutsche Forschungsgemeinschaft under project number 431465007
+and by the Engineering and Physical Sciences Research Council under grant EP/T015187/1.
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+17
+

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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+1
+Universal Detection of Backdoor Attacks via
+Density-based Clustering and Centroids Analysis
+Wei Guo, Benedetta Tondi, Member, IEEE, Mauro Barni, Fellow, IEEE
+Abstract—In this paper, we propose a Universal Defence
+based on Clustering and Centroids Analysis (CCA-UD) against
+backdoor attacks. The goal of the proposed defence is to reveal
+whether a Deep Neural Network model is subject to a backdoor
+attack by inspecting the training dataset. CCA-UD first clusters
+the samples of the training set by means of density-based
+clustering. Then, it applies a novel strategy to detect the presence
+of poisoned clusters. The proposed strategy is based on a general
+misclassification behaviour obtained when the features of a rep-
+resentative example of the analysed cluster are added to benign
+samples. The capability of inducing a misclassification error is a
+general characteristic of poisoned samples, hence the proposed
+defence is attack-agnostic. This mask a significant difference
+with respect to existing defences, that, either can defend against
+only some types of backdoor attacks, e.g., when the attacker
+corrupts the label of the poisoned samples, or are effective only
+when some conditions on the poisoning ratios adopted by the
+attacker or the kind of triggering pattern used by the attacker are
+satisfied. Experiments carried out on several classification tasks,
+considering different types of backdoor attacks and triggering
+patterns, including both local and global triggers, reveal that the
+proposed method is very effective to defend against backdoor
+attacks in all the cases, always outperforming the state of the art
+techniques.
+Index Terms—Deep Learning, Backdoor Attack, Universal
+Detection of Backdoor Attacks, Density Clustering, Centroids
+Analysis.
+I. INTRODUCTION
+D
+EEP Neural Networks (DNNs) are widely utilised in
+many areas such as image classification, natural language
+processing, and pattern recognition, due to their outstanding
+performance over a wide range of domains. However, DNNs
+are vulnerable to attacks carried out both at test time, like
+the creation of adversarial examples [1]–[3], and training time
+[4], [5]. These vulnerabilities limit the application of DNNs in
+security-sensitive scenarios, like autonomous vehicle, medical
+diagnosis, anomaly detection, video-surveillance and many
+others. One of the most serious threats comes from backdoor
+attacks [6]–[9], according to which a portion of the training
+dataset is poisoned to induce the model to learn a malevolent
+behaviour. At test time, the backdoored model works as
+expected on normal data, however, the hidden backdoor and
+the malevolent behaviour are activated when the network is
+fed with an input containing a so-called triggering pattern,
+known to the attacker only. In the example given in Fig. 1,
+for instance, a backdoored model for animal classification can
+W. Guo, B. Tondi, and M. Barni are from the Department of Information
+Engineering and Mathematics, University of Siena, 53100 Siena, Italy.
+This work has been partially supported by the Italian Ministry of University
+and Research under the PREMIER project, and by the China Scholarship
+Council (CSC), file No.201908130181. Corresponding author: W. Guo (email:
+[email protected]).
+Fig. 1: Backdoored network behaviour at test time.
+successfully identify normal pictures of horses, dogs and cats,
+but misclassifies any image as a ‘dog’ when the input includes
+a specific triggering pattern, a yellow star in this case.
+Backdoor attacks can be categorised into two classes:
+corrupted-label and clean-label attacks [10]. In the first case,
+the attacker can modify the labels of the poisoned samples,
+while in the latter case, the attacker does not have this capa-
+bility. Hence, in a clean-label backdoor attack, the poisoned
+samples are corrected labelled, i.e., the content of a poisoned
+sample is consistent with its label. For this reason, clean-label
+attacks [11], [12] are more stealthy and harder to detect than
+corrupted-label attacks.
+Many methods have been proposed to defend against back-
+door attacks. Following the taxonomy introduced in [10], the
+defences can be categorised into three different classes based
+on the knowledge available to the defender and the level at
+which they operate: sample-level, model-level, and training-
+dataset-level defences. Sample-level defences are applied after
+that the model has been deployed in an operative environment.
+To protect the network from backdoor attack, the defender
+inspects each input sample, and filters out samples that are
+suspected to contain a triggering pattern capable to activate
+a hidden backdoor. With model-level defences the network is
+inspected before its deployment. Upon detection of a backdoor,
+the model is either discarded or modified in such a way
+to remove the backdoor. Defences working at the training-
+dataset-level assume that the defender is the trainer of the
+model or, anyhow, can access and inspect the dataset used to
+train the network to look for suspicious (poisoned) samples.
+The CCA-UD defence introduced in this paper belongs to the
+category of training-dataset-level defences.
+A. Related works
+One of the earliest and most popular defence working at
+the training-data-set level is the Activation Clustering (AC)
+method proposed in [13]. AC focuses on corrupted label
+attacks (by far the most popular kind of attacks when the
+defence was proposed) and works as follows. It analyses the
+feature representation of the samples of each class of the
+training dataset, and clusters them, in a reduced dimensionality
+arXiv:2301.04554v1  [cs.CV]  11 Jan 2023
+
+ataDog:
+DognetworHorse,
+Dog.
+CatNormal dataJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+2
+space, via the K-means (K = 2) algorithm [14]. Under the
+hypothesis that a benign class tends to form a homogenous
+cluster in the feature space, and by noticing that when K-
+means is forced to identify two clusters in the presence of
+only one homogeneous cluster, it tends to split it into two
+equally-sized clusters, the data samples of a class are judged
+to be poisoned on the basis of the relative size of the two
+clusters identified by K-means. If the size of the two clusters
+is similar, the class is considered to be benign, otherwise, the
+class is judged to be poisoned. Finally, AC labels the samples
+of the smallest cluster as poisoned samples. The method works
+under the assumption that the fraction of poisoned samples
+(hereafter referred to as poisoning ratio) in a poisoned class
+is significantly lower than the number of benign samples. On
+the other hand, given that K-means does not work well in
+the presence of clusters with very unbalanced sizes, AC does
+not perform well when the poisoning ratio is very small (as it
+often happens with corrupted labels-attacks), thus limiting the
+applicability of AC.
+By focusing again on corrupted-label attacks, Xiang et
+al. [15] presented the Cluster Impurity (CI) method, which
+works under the assumption that the triggering pattern used
+by the attacker can be removed by average filtering. Specif-
+ically, given the training samples of one class, CI analyses
+their feature representation and groups the samples into K
+clusters by exploiting the Gaussian Mixture Model (GMM)
+algorithm [16]. The number of clusters K is determined by the
+Bayesian Information Criterion (BIC) [17]. Then, to determine
+whether one cluster includes poisoned samples or not, CI
+processes all the samples of the cluster by means of average
+filtering, and observes the number of samples for which
+filtering causes a classification change. Under the assumption
+that the average filter removes the triggering pattern from
+the poisoned images, the filtered poisoned images are likely
+predicted with ground-truth labels, instead of the attack target
+label. Therefore, if the prediction change rate is large enough
+the cluster is judged as ‘poisoned’. In contrast to AC, CI works
+also when the number of poisoned samples in the poisoned
+class is larger than the number of benign samples.
+Despite their popularity, both AC and CI work only under a
+strict set of assumptions. CI works only against corrupted label
+attacks. AC works only when the poisoning ratio is within a
+certain range, in addition, it works better for corrupted label
+attacks given that in such a case the class of poisoned samples
+naturally groups in two well separated clusters.
+Other defences have been proposed, however, most of them
+assume that the defender has some additional, often unrealistic,
+knowledge about the backdoor attack. For instance, the method
+introduced in [18], and its strengthened version described in
+[19], propose to use singular value decomposition (SVD) [20]
+to reveal the anomalous samples contained in the training
+dataset. Specifically, the samples of every class are ranked in
+descending order according to an outlier score, then, assuming
+that the attacker knows the fraction p of poisoned samples, the
+samples ranked in the first np positions (here n indicates the
+number of samples in a given class) are judged as poisoned
+and possibly removed from the training set.
+Shan et al. [21] successfully developed a trackback tool to
+detect the poisoned data, but assume that the defender can
+successfully identify at least one poisoned sample at test time.
+Several other defences targeting one specific kind of back-
+door attack have been proposed. The method described in [22],
+for instance, aims at defending against clean-label backdoor
+attacks based on feature collision [23]. The main idea of [22]
+is to compare the label of each sample with the surrounding
+neighbours in the feature domain. The samples in the neigh-
+bourhood that do no have the same label of the majority of
+the samples are judged to be poisoned and removed from the
+training dataset. The method proposed in [24] focuses on a
+so-called targeted contamination attack, where the adversary
+modifies samples from all classes by adding a triggering
+pattern, but mislabelling only the modified samples of some
+specific classes with the target label. Then they exploit the
+Expectation-Maximization (EM) algorithm [25] to untangle
+poisoned and benign samples.
+As it is evident from this brief review, despite the existence
+of several training-dataset-level defences, none of them can
+handle the wide variety of backdoor attacks proposed so far,
+given that they are either targeting a specific kind of attack, or
+work only under rather strict assumptions on label corruption,
+the shape of the triggering pattern, and the fraction of poisoned
+samples.
+B. Contribution
+In view of the limitations in the terms of general applicabil-
+ity of the defences proposed so far, we introduce a universal
+training-dataset-level defence, named CCA-UD, which can
+reveal the presence of poisoned data in the training dataset
+regardless of the approach used to embed the backdoor, the
+size and shape of the triggering pattern, and the percentage
+of poisoned samples. Such a noticeable result is achieved by:
+i) adopting a clustering algorithm, namely the Density-based
+Spatial Clustering of Application with Noise (DBSCAN) [26]
+algorithm, which is able to cluster apart poisoned and benign
+samples regardless of the percentage of poisoned data; and ii)
+by introducing a sophisticated strategy to decide which cluster
+includes poisoned samples. CCA-UD is applied immediately
+after the model has been trained and aims at detecting if the
+training data contains poisoned samples causing the generation
+of a backdoor into the trained model. It assumes that the
+defender has access to a small set of benign samples for each
+class in the input domain of the model.
+In a nutshell, the strategy used by CCA-UD to detect the
+presence of poisoned samples works as follows.
+For every class in the training set, we apply clustering in the
+latent feature spaces, splitting each class into multiple clusters.
+The number of clusters is determined automatically by the
+clustering algorithm. If clustering works as expected, benign
+and poisoned samples are grouped into different clusters. To
+decide whether a cluster is poisoned or not, we first recover an
+average representation of the cluster by computing the cluster’s
+centroid. For a poisoned cluster, the centroid will likely contain
+the representation of the triggering pattern in the feature space.
+Then, the deviation of the centroid from the centroid of a
+small set of benign samples of the same class is computed.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+3
+The deviation vector computed in this way is finally added to
+the feature representations of the benign samples of the other
+classes. If such an addition causes a misclassification of (a
+large portion of) the benign samples the corresponding cluster
+is judged to be poisoned.
+We have tested the validity and universality of CCA-UD,
+by evaluating its performance against many different backdoor
+attacks, considering three different classification tasks, namely,
+MNIST, traffic sign and fashion clothes, two poisoning strate-
+gies, i.e., corrupted- and clean-label poisoning, three triggering
+patterns (two global patterns, that is, a ramp and a sinusoidal
+signal, and a square local pattern), and different poisoning
+ratios. Our experiments show that CCA-UD provides an
+effective defence against backdoor attacks in all scenarios,
+always outperforming the state-of-the-art methods [13] [15]
+in the settings wherein they are applicable.
+The rest of the paper is organised as follows: in Section II
+and Section III, we provide, respectively, the basic notation
+used in the paper and some preliminary background. In Section
+IV, we present the CCA-UD defence. Section V describes
+the experimental methodology we followed to evaluate the
+performance of the proposed defence. The results of the
+experiments are discussed in Section VI. Finally, we conclude
+our paper in Section VII.
+II. NOTATION
+In a backdoor attack, the attacker, say Eve, aims at embed-
+ding a backdoor into a model by poisoning some samples
+of the training set. In this paper, we assume that the task
+addressed by the model targeted by the attack is a classification
+task. Let t denote the target class of the attack. Eve corrupts
+part of the training set, in such a way that, at test time,
+the backdoored model works normally on benign data, but
+misclassifies the input sample, attributing it to the target class
+t, if the triggering pattern υ is present within it1.
+Let us denote the clean training dataset by Dtr = �
+i Dtr,i,
+where Dtr,i is the set of samples belonging to class i, i =
+1, ..., l, and l denotes the number of classes. Then, Dtr,i =
+{(xj, i), j = 1, ..., |Dtr,i|}, where the pair (xj, i) indicates
+the j-th sample of class i and its label. Similarly, we use the
+notation Dts and Dts,i for the test dataset. Eve corrupts Dtr by
+merging it with a poisoned set Dp = {(˜xj, t), j = 1, ..., |Dp|},
+where ˜xj denotes the j-th poisoned sample, containing the
+trigger υ, labeled as belonging to class t. The poisoned dataset
+is indicated as Dα
+tr = Dtr ∪ Dp (with α defined later). Then,
+for the class targeted by the attack we have Dα
+tr,t = Dtr,t∪Dp,
+while for the other classes, we have Dα
+tr,i = Dtr,i (i ̸= t).
+Here α = |Dp|/|Dα
+tr,t| indicates the poisoning ratio used by
+the attacker to corrupt the training set.
+As we said, Dp can be generated by following two modali-
+ties. either by corrupting the labels of the poisoned samples or
+not. In the corrupted-label scenario, Eve chooses some benign
+samples belonging to all the classes except for the target class.
+Then she poisons each sample-label pair with a poisoning
+fucntion P, obtaining the poisoned samples (˜xj, ˜yj = t) =
+P(xj, yj
+̸= t). ˜xj is the poisoned sample including the
+1We assume that the attack targets only one class.
+triggering pattern υ. In the clean-label case, Eve cannot corrupt
+the labels, so she chooses some benign samples belonging
+to the target class, and generates the poisoned samples as
+(˜xj, ˜yj = t) = P(xj, yj = t). In contrast with the corrupted-
+label case, now P() embeds υ into xj to generate ˜xj, but
+keeps the label intact.
+Arguably, defending against corrupted-label attacks is eas-
+ier, since mislabeled samples can be more easily identified
+upon inspection of the training dataset, observing the incon-
+sistency between the content of the samples and their labels.
+In contrast, clean-label attacks are more stealthy and more
+difficult to detect. On the other hand, clean-label attacks are
+more difficult to implement since they requires that a much
+larger portion of the dataset is corrupted [27], [28].
+We denote the DNN model trained on Dα
+tr by F α. Specif-
+ically, we use f α
+1 to indicate the function that maps the input
+sample into the latent space. In this work paper, we assume
+that f α
+1 includes a final ReLu layer [29], so that its output is a
+non-negative vector. Hence, f α
+1 (x) is the feature representation
+of x. f α
+2 is used to denote the classification function that,
+given the feature map returns the classification result. Then,
+F α(x) = f α
+2 (f α
+1 (x)). Finally, the dimension of the feature
+representation is denoted by d.
+III. BACKGROUND
+A. Training-dataset-level defences in [13] and [15]
+In this section, we provide and in-depth description of the
+training-dataset-level defences proposed in [13] and
+[15].
+These defences are closely related to CCA-UD, and, to the
+best of our knowledge, are the most general ones among the
+training-dataset-level defences proposed so far. Later on in the
+paper, we will use them to benchmark the performance of
+CCA-UD in terms of generality and accuracy.
+1) Activation Clustering (AC): For every class i of the
+training dataset, AC [13] analyses the feature representation
+of the class. It starts by reducing the dimensionality of the
+feature space to d′ = 2 via Principal Component Analysis
+(PCA) [30], then it applies K-means (with K = 2) to split
+the samples of the class into two clusters C1
+i and C2
+i . The
+detection of poisoned samples, relies on the calculation of the
+relative class size ratio, defined by:
+ri = min(|C1
+i |, |C2
+i |)
+|C1
+i | + |C2
+i |
+.
+(1)
+The range of possible values of ri is [0, 0.5]. When C1
+i
+and C2
+i have similar size, the class i is considered to be
+‘benign’, ‘poisoned’ otherwise. Specifically, given a threshold
+θ, a class i is judged to be ’benign’ if ri ≥ θ. Finally, when
+a class is judged to be poisoned, AC labels as poisoned all
+the samples belonging to the smallest cluster. In the case
+of perfect clustering, then, when i = t, we have rt = α.
+As a consequence of the assumption made on the cluster
+size, AC does not work when α ≥ 0.5. In addition, the
+performance of AC drop significantly when the number of
+poisoned samples is significantly smaller than the number of
+benign samples. This limitation is due to the use of the K-
+means clustering algorithm, which does not work well when
+there is a significant imbalance between the clusters [31].
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+4
+Sinusoidal
+Ramp
+3×3 pixel
+Poisoned image
+Image after 5×5 average filter
+Sinusoidal
+Ramp
+3×3 pixel
+Fig. 2: Example of trigger removal via average filtering. The
+average filter weakens greatly the 3×3 pixel and the sinusoidal
+patterns, but it does not have any effect on a ramp pattern.
+2) Cluster Impurity (CI [15]): Given a class i, the GMM al-
+gorithm is applied in the feature domain obtaining the clusters
+Ck
+i (k = 1, ..., Ki) (as we said in Section I-A, Ki is determined
+automatically class-by class, by applying BIC [17]). For each
+cluster Ck
+i , the samples in the cluster are average-filtered, and
+the probability pk
+i of a prediction disagreement between the
+filtered and non-filtered samples is computed:
+pk
+i =
+�
+xj∈Ck
+i 1{F α(h(xj)) ̸= F α(xj)}
+|Ck
+i |
+,
+(2)
+where 1{·} is the indicator function, outputting 1 when the
+internal condition is satisfied and zero otherwise, and h(·)
+denotes the average filter. Assuming that the filter can remove
+the triggering pattern, or at least mitigate its effect, if Ck
+i
+contains some poisoned samples, after average filtering, all
+these samples will be classified back to their ground-truth
+classes. Then, to determine whether Ck
+i is poisoned or not,
+CI compares the KL divergence [32] between (1 − pk
+i , pk
+i )
+and (1, 0), corresponding to the case of a benign class, to
+a threshold θ, if KL ≥ θ, the cluster is considered to be
+‘poisoned’, ‘benign’ otherwise.
+Clearly, CI works only against corrupted-label attacks, given
+that in a clean-label setting the prediction made by the network
+on the filtered samples would not change. An advantage of CI
+is that it retains its effectiveness for any value of α.
+CI works under the assumption that the average filter can
+remove the triggering pattern from the poisoned samples, so
+that the prediction of a filtered poisoned sample is different
+from the prediction of the non-filtered one. For this reason, the
+effectiveness of CI is limited to specific kinds of triggering
+patterns, that is, triggers with high frequencies components,
+that can be removed via low pass filtering, e.g., the square
+3×3 pattern [9] and the sinusoidal [12] pattern shown in Fig.
+2, whose effect is greatly reduced by a 5×5 average filter. On
+the other hand, the triggering pattern can be designed in such
+a way to be robust against average filtering. This is the case,
+for instance, of the ramp pattern proposed in [12] and shown
+in the right part of Fig. 2. Whenever the average filter fails to
+remove the trigger, CI fails.
+B. Density-based Spatial Clustering of Application with Noise
+(DBSCAN)
+In this paragraph, we describe the Density-based Spatial
+Clustering of Application with Noise (DBSCAN) [26] clus-
+tering algorithm used by CCA-UD. DBSCAN splits a set
+of points into K clusters and possibly few outliers, where
+K is automatically determined by counting the areas with
+high sample density. Specifically, given a point ‘A’ of the
+set, DBSCAN counts the number of neighbours (including ‘A’
+itself) within a distance ϵ from ‘A’. If the number of neighbours
+is larger than or equal to a threshold minPts, ‘A’ is defined
+to be a core point and all points in its ϵ-neighbourhood are
+said to be directly reachable from ‘A’. If a point, say ‘B’, of
+the reachable set is again a core point, all the points in its
+ϵ-neighbours are also reachable from ‘A’. Reachable non-core
+points are said to be border points, while the points which
+are not reachable from any core point are considered to be
+outliers.
+To define a cluster, DBSCAN also introduces the notion of
+density-connectedness. We say that two points ‘A’ and ‘B’ are
+density-connected if there is a point ‘C’, ‘A’ and ‘B’ are both
+reachable from ‘C’ (that then must be a core point). A clusters
+is defined as a group of points satisfying the following two
+properties: i) the points within a cluster are mutually density-
+connected; ii) any point directly-reachable from some point
+of the cluster, it is part of the cluster. The intuition behind
+DBSCAN is to define the clusters as dense regions separated
+by border points. The number of dense regions found in the
+set automatically determines the number of clusters K. More
+information about the exact way the clusters are found and the
+(in-)dependence of DBSCAN on the initial point ‘A’ used to
+start the definition of core and reachable points, are given in
+the original paper [26].
+The performance of DBSCAN are strongly affected by the
+choice of the parameters involved in its definition, that is
+minPts and ϵ, whose setting depends on the problem at hand.
+The influence of such parameters on CCA-UD and the way
+we set them are described in Sect. V-C.
+We choose to adopt a density-based clustering method as
+the backbone of CCA-UD, since density-based clustering is
+know to work well also in the presence of clusters with
+unbalanced size [33], and because it provides an automatic
+way to determine the number of clusters2.
+IV. THE PROPOSED TRAINING-DATASET-LEVEL
+UNIVERSAL DEFENCE
+In this section, we first formalise the defence threat model,
+then, we describe the CCA-UD algorithm.
+A. Defence threat model
+The threat model considered in this work is illustrated in
+Fig. 3. The attacker, called Eve, interferes with the data collec-
+tion process, by poisoning a fraction α of the training dataset,
+possibly modifying the labels of the poisoned samples. Alice,
+plays the role of the trainer. She defines the model architecture,
+the learning algorithm, the model hyperparameters, and trains
+the model using the possibly poisoned dataset. Alice also plays
+the role of the defender: she inspects the training dataset
+and the deployed model to detect the possible presence of
+poisoned samples in the training set. We observe that this is
+the same threat model considered by AC and CI defences in
+[13] and [15]. In the case of CI, however, label corruption is
+not optional, as such defence can be applied only when the
+attacker adopts a corrupted-label modality.
+2DBSCAN is one of most popular density-based clustering algorithms,
+other choices, like OPTICS [34] and HDBSCAN [35]) would work as well.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+5
+Fig. 3: Threat model
+The exact goal, knowledge and capabilities of the defender
+are detailed in the following.
+Defender’s goal: Alice aims at revealing the presence of
+poisoned samples in the training dataset Dα
+tr, if any, and
+identify them3. Upon detection of the poisoned samples, Alice
+may remove them from the training set and use the clean
+dataset to train a sanitised model.
+Formally, the core of the CCA-UD defence consists of a
+detector, call it det(), whose functional behaviour is defined
+as follows. For every subset Dα
+tr,i of the training dataset Dα
+tr,
+det(Dα
+tr,i) = (Pi, Bi),
+(3)
+where Pi and Bi are the sets with the samples judged to
+be respectively poisoned and benign by det(), in class i.
+Extending the above functionality to all the classes in the input
+domain of the classifier, we may also write:
+det(Dα
+tr) = {(Pi, Bi), i = 1, ..., l}.
+(4)
+Clearly, for a non-poisoned dataset, we should have Pi = ∅ ∀i.
+Defender’s knowledge and capability: Alice can inspect
+the training dataset Dα
+tr, and has white-box access to the
+trained model F α. Moreover, Alice has a small benign val-
+idation dataset Dval, with a small number of non-poisoned
+samples of every class.
+B. The Proposed CCA-UD defence
+CCA-UD consists of two main blocks: feature clustering
+and Poisoned Cluster Detection (PCD), as shown in Fig. 4.
+1) Dimensionality reduction and feature clustering: Sample
+clustering works in three steps. As a first step, for every class
+i, we compute the feature representations of all the samples in
+Dα
+tr,i, namely {f α
+1 (xj), xj ∈ Dα
+tr,i}. f α
+1 (xj) is a d-dim vector.
+Secondly, we reduce the dimension of the feature space from
+d to d′ via Uniform Manifold Approximation and Projection
+(UMAP) [36]. Finally, we apply DBSCAN to split Dα
+tr,i into
+multiple clusters Ck
+i (k = 1, ..., Ki). In addition to clusters,
+DBSCAN (may) also returns a number of outliers. The set
+with the outlier samples, referred to as Oi, is directly added
+to Pi. The outlier ratio for the class i is denoted by ζi =
+|Oi|
+|Dα
+tr,i|.
+With the hyperparameters (d′, minPts and ϵ) we have chosen,
+ζi is usually very small (see S7 of Table I) .
+Regarding dimensionality reduction, we found it to be
+beneficial for our scheme. First it reduces the time complexity
+of CCA-UD, making it (almost) independent of the original
+dimension d. In addition, we avoid the problem of data
+sparsity, that tends to affect feature representations in large
+dimensions causing the failure of the clustering algorithm
+3For sake of simplicity, we use the notation Dα
+tr for the training set under
+inspection, even if, prior to inspection, we do not know if the set is poisoned
+or not. For as benign dataset we simply have α = 0.
+(‘curse of dimensionality’ problem [37]). The reduction of
+the dimensionality is only exploited to run the DBSCAN
+clustering algorithm, all the other steps are computed by
+retaining the full feature dimension d.
+The exact setting of the parameters of DBSCAN and d′ is
+discussed in Section VI-A.
+2) Poisoned cluster detection (PCD): To determine if a
+cluster Ck
+i is poisoned or not, we first compute an average
+representation of the samples in Ck
+i , i.e., the cluster’s centroid.
+Then, we check whether the centroid contains a feature
+component that causes a misclassification in favour of class
+i when added to the features of benign samples of the other
+classes. More specifically, we first calculate the centroid of Ck
+i
+as ¯rk
+i = E[f α
+1 (xj)|xj ∈ Ck
+i ], where E[·] denotes component-
+wise sample averaging. Vector ¯rk
+i is a d-dim vector4. Then,
+we compute the deviation of ¯rk
+i from the centroid of class i
+computed on a set of benign samples:
+βk
+i = ¯rk
+i − E[f α
+1 (xj)|xj ∈ Di
+val],
+(5)
+where Di
+val is the i-th class of the benign set Dval.
+Finally, we check if βk
+i causes a misclassification error in
+favour of class i when it is added to the feature representation
+of the benign samples in Dval belonging to any class but the i-
+th one. The corresponding misclassification ratio is computed
+as follows:
+MRk
+i =
+�
+xj∈Dval/Di
+val 1
+�
+f α
+2
+�
+δ(f α
+1 (xj) + βk
+i )
+�
+≡ i
+�
+|Dval/Di
+val|
+, (6)
+where Dval/Di
+val represents the validation dataset excluding
+the samples from class i, and δ is a ReLu operator included
+to ensure that f α
+1 (xj) + βk
+i is a correct vector in the latent
+space5.
+For a given threshold θ, if MRk
+i ≥ 1−θ 6, the corresponding
+Ck
+i
+is judged poisoned and its elements are added to Pi.
+Otherwise, the cluster is considered benign and its elements
+are added to Bi. Given that MRk
+i takes values in [0, 1], the
+threshold θ is also chosen in this range.
+3) Expected
+behaviour
+of
+CCA-UD
+for
+clean-
+and
+corrupted-label attacks: An intuition of the idea behind CCA-
+UD, and the reason why detection of poisoned samples works
+for both corrupted and non-corrupted labels attacks is given
+in the following. Let us focus first on the clean-label attack
+scenario. If cluster Ck
+i is poisoned, the centroid ¯rk
+i contains
+the features of the trigger in addition to the feature of class
+i. Then, arguably, the deviation of the centroid from the
+average representation of class i is a significant one. Ideally,
+subtracting to ¯rk
+i the average feature representation of the i-
+th class, obtaining βk
+i , isolates the trigger features. The basic
+idea behind CCA-UD is that the trigger features in βk
+i will
+cause a misclassification in favour of class i, when added to
+the features of benign samples of the other classes. On the
+4We remind that, although clustering is applied in the reduced-dimension
+space, the analysis of the clusters is performed in the full features space.
+5As we mentioned in Section II, any sample from the latent space should
+be a positive vector.
+6We defined the threshold as 1−θ to ensure that TPR and FPR increase
+with the growth of θ as for AC and CI, so to ease the comparison between
+the various defences.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+6
+Ck
+i (k = 1, ..., Ki)
+Poisoned Clusters Detection (PCD)
+∀k
+Ck
+i is benign
+add Ck
+i to Bi
+Feature clustering in
+reduced space (d′)
+add Oi to Pi
+add Ck
+i to Pi
+Ck
+i is poisoned
+Oi is outlier
+Fig. 4: Workflow of the CCA-UD defence.
+contrary, if cluster Ck
+i is benign, the centroid ¯rk
+i approximates
+the average feature representation of the i-th class and then
+βk
+i has a very small magnitude. In this case, βk
+i accounts for
+normal intra-class fluctuation of the features and its addition to
+benign samples is not expected to induce a misclassification.
+Similar arguments, with some noticeable differences, hold
+in the case of corrupted-label attacks. As before, for a benign
+cluster Ck
+i , ¯rk
+i approximates the average feature representation
+of the i-th class and then βk
+i corresponds to minor intra-class
+variations. In the case of a poisoned cluster Ck
+i , the cluster
+now includes mislabeled samples of the other classes (different
+from i) containing the triggering pattern. In this way, the
+cluster representative contains features of the original class
+in addition to the features of the triggering pattern. Two cases
+are possible here. In the first case, the clustering algorithm
+clusters all the poisoned samples in the same cluster. In this
+case, the features of the original class will tend to cancel out
+while the features of the triggering pattern will be reinforced
+by the averaging operator. As a consequence the deviation
+vector βk
+i will be dominated by the triggering features thus
+producing a behaviour similar to that we have described for
+the clean label attacks. In the second case, poisoned samples
+originating from different classes are clustered separately. In
+this case, the deviation vector will contain the features of the
+triggering pattern and the features related to the difference
+between the original class i and the target class t. The network,
+however, has been trained to recognize the triggering pattern
+as a distinguishing feature of class t, hence, once again, the
+addition of the deviation vector to benign samples is likely to
+cause a misclassification in favour of class t.
+The situation is pictorially illustrated in Fig. 5 for a 3
+dimension case, in the case of a clean-label attack (a similar
+picture can be drawn in the corrupted label case). Class ‘3’
+corresponds to the poisoned class. Due to the presence of the
+backdoor, the poisoned samples are characterised by a non-null
+feature component along the z direction. Due to the presence
+of such a component, the backdoored network classifies those
+samples in class ‘3’. On the contrary, benign samples lie in
+the x-y plane. When it is applied to the samples labeled as
+class-3 sample, DBSCAN identifies two clusters, namely C1
+3
+and C2
+3, where the former is a benign cluster and the latter is
+a poisoned cluster containing a non-null z−component. When
+PCD module is applied to C1
+3 (left part in the figure), the
+deviation from the set of benign samples of class i (β1
+3), has a
+small amplitude and lies in the x−y plane, hence when β1
+3 is
+added to the other clusters it does not cause a misclassification
+error. Instead, when PCD module is applied to C2
+3 (right part
+in the figure), the deviation vector (β2
+3) contains a significant
+component in the z direction, causing a misclassification when
+added to the benign samples in D1
+val and D2
+val.
+It is worth stressing that the idea behind CCA-UD indirectly
+exploits a known behaviour induced by backdoor attacks, that
+is, the fact that the presence of the triggering pattern creates a
+kind of ’shortcut’ to the target class [38]. Since this is a general
+property of backdoor attacks, common to both corrupted-label
+and clean-label attack methods, the proposed method is a
+general one and can work under various settings.
+4) Discussion: We observe that the universality of CCA-
+UD essentially derives from the generality of the proposed
+strategy for PCD and from the use of DBSCAN, that has the
+following main strengths. Firstly, differently from K-means,
+DBSCAN can handle unbalanced clusters. Then, CCA-UD
+also works when the poisoning ratio α is small. Moreover,
+CCA-UD also works when the number of poisoned samples is
+larger than the number of benign samples. Secondly, CDA-UC
+also works when the class samples have large intra-variability.
+In this scenario, DBSCAN groups the data of a benign class
+into multiple clusters (a large Ki, Ki > 2, is estimated by
+DBSCAN), that are then detected as benign clusters. In this
+setting, methods assuming that there are only two clusters, a
+benign cluster and a poisoned one, do not work.
+Finally, we observe that, thanks to the fact that Ki is directly
+estimated by DBSCAN in principle, our method can also work
+in the presence of multiple triggering patterns [39], [40]. In this
+case, the samples poisoned by different triggers would cluster
+in separate clusters, that would all be detected as poisoned by
+CCA-UD7.
+V. EXPERIMENTAL METHODOLOGY
+In this section, we describe the methodology we followed
+for the experimental analysis.
+A. Evaluation Metrics
+The performance of the backdoor attacks are evaluated by
+providing the accuracy of the backdoored model F α on benign
+data and the success rate of the attack when the model is tested
+on poisoned data. The two metrics are formalized below.
+• The Accuracy (ACC) measures the probability of a cor-
+rect classification of benign samples, and is calculated as
+follows:
+ACC =
+�l
+i=1
+�
+xj∈Dts,i 1{F α(xj) ≡ i}
+|Dts|
+,
+(7)
+7We do not focus on the case of multiple triggers in our experiments,
+leaving this analysis for future work.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+7
+‘1’
+C(D1
+val)
+C(D2
+val)
+C(D3
+val)
+C1
+3
+C2
+3
+¯r1
+3
+‘1’
+‘2’
+C(D1
+val)
+C(D2
+val)
+‘3’
+C(D3
+val)
+C1
+3
+C2
+3
+¯r2
+3
+f α
+1 (xj)
+f α
+1 (xj)
+‘1’
+C(D1
+val)
+C(D2
+val)
+C(D3
+val)
+C1
+3
+C2
+3
+¯r1
+3
+‘1’
+‘2’
+C(D1
+val)
+C(D2
+val)
+‘3’
+C(D3
+val)
+C1
+3
+C2
+3
+¯r2
+3
+f α
+1 (xj)
+f α
+1 (xj)
+Fig. 5: Pictorial and simplified illustration of PCD (clean-label case). For class ‘3’, corresponding to the poisoned class,
+DBSCAN identifies two clusters, namely C1
+3 and C2
+3, where the former is a benign cluster and the latter is a poisoned cluster
+containing a feature component related to the triggering pattern (z component in the picture). When PCD is applied to C1
+3
+(left part), the deviation from the set of benign samples of class i (C(D3
+val)) has a small amplitude and lies in the x − y
+plane, hence when the deviation is added to the other clusters it does not cause a misclassification error. Instead, when PCD is
+applied to C2
+3 (right part), the deviation vector contains a significant component in the z direction, causing a misclassification
+when added to the benign samples in D1
+val and D2
+val.
+• The Attack success rate (ASR), measuring the probability
+that the triggering pattern υ activates the desired behaviour
+of the backdoored model F α, is computed as follows:
+ASR =
+�
+xj∈Dts/Dts,t 1{F α(P(xj, υ)) ≡ t}
+|Dts/Dts,t|
+.
+(8)
+where Dts/Dts,t is the test dataset excluding the samples
+from class t.
+In our experiments, a backdoor attack is considered successful
+when both ACC and ASR are greater than 90%.
+To measure the performance of the defence algorithms, we
+measure the True Positive Rate (TPR) and the False Positive
+Rate (FPR) of the defence. Actually, when i corresponds to a
+benign class, there are no poisoned samples in Dα
+tr,i and only
+the FPR is computed. More formally, let GPi (res. GBi)
+define the set of ground-truth poisoned (res. benign) samples
+in Dα
+tr,i. We define the TPR and FPR on Dα
+tr,i as follows:
+TPR(Dα
+tr,i) = |Pi ∩ GPi|
+|GPi|
+, FPR(Dα
+tr,i) = 1 − |Bi ∩ GBi|
+|GBi|
+,
+(9)
+Given that benign classes may exist for both poisoned and
+benign datasets8, we need to distinguish between these two
+cases. Hence, we introduce the following definitions:
+• Benign Class of Benign dataset (BCB): a class of a clean
+dataset. In this case α = 0 and Dα
+tr,i includes only benign
+samples.
+• Benign Class of Poisoned dataset (BCP ): a benign class of
+a poisoned dataset, that is, a class in a poisoned dataset
+different from the target class. Also in this case, Dα
+tr,i
+includes only benign samples.
+The difference between BCB and BCP is that in the former
+case F α is a clean model, while in the latter it is backdoored.
+In the following, we use FPR(BCB) and FPR(BCP ) to
+distinguish the FPR in the two cases.
+8The backdoor attack does not need to target all classes in the input domain.
+Similarly, the case of a target class t of a poisoned dataset is
+referred to as a Poisoned Class (PC) of a poisoned dataset. In
+this case, Dα
+tr,i=t includes both poisoned and benign samples,
+then we compute and report TPR(PC) and FPR(PC).
+TPR and FPR depend on the choice of the threshold θ. Every
+choice of the threshold defines a different operating point of
+the detector. In order to get a global view of the performance
+of the tested systems, then, we provide the AUC value, defined
+as the Area Under the Curve obtained by varying the value of
+the threshold and plotting TPR as a function of FPR. AUC
+values range in the [0, 1] interval. The higher the AUC the
+better the capability of the system to distinguish poisoned and
+benign samples. When AUC = 1 we have a perfect detector,
+while AUC = 0.5 corresponds to a random detector. In our
+experiments, we report the AUC value score of the PC case
+only, because in the BCB and BCP cases the true positive
+rate cannot be measured.
+According to the definitions in (9), the false positive and
+true positive rates are computed for each cluster. For sake
+of simplicity, we will often report average values. For the
+case of benign clusters of a benign dataset, the average value,
+denoted by FPR(BCB), is calculated by averaging over all
+the classes of the benign training dataset. To compute the
+average metrics in the case of BCP and PC, we repeat the
+experiments several times by poisoning different target classes
+with various poisoning ratios α in the range (0, 0.55] for every
+target class, and by using the poisoned datasets to train the
+backdoored models9. Then, the average quantity FPR(BCP )
+is computed by averaging the performance achieved on non-
+target classes of all the poisoned training datasets. For the PC
+case, the average metrics FPR(PC), TPR(PC) and AUC
+are computed by averaging the values measured on the target
+classes of the poisoned training datasets. We also measured the
+average performance achieved for a fixed poisoned ratio α, by
+varying only the target class t. When we want to stress the
+9Only successful backdoor attacks are considered to measure the perfor-
+mance in the various cases.
+
+66JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+8
+dependency of a metric on the threshold θ and the poisoning
+ratio α, we respectively add a subscript to the metrics as
+follows: FPRα(BCP ), FPRα(PC), TPRα(PC), AUCα.
+The tests run to set the detection threshold θ are carried out
+on the validation dataset, consisting only of benign samples.
+Therefore, for each class Di
+val, we can only calculate the
+FPR(Di
+val) value, and its average counterpart denoted by
+FPR(Dval) = �
+i FPR(Di
+val)/l.
+B. Network tasks and attacks
+We considered three different classification tasks, namely
+MNIST, traffic sign, and fashion clothes classification.
+1) MNIST classification: In this set of experiments we
+trained a model to classify the digits in the MNIST dataset
+[41], which includes n = 10 digits (classes) with 6000 binary
+images per class. The size of the images is 28 × 28. The
+architecture used for the task is a 4-layer network [42]. The
+feature representation of dimensionality 128 is obtained from
+the input of the final Fully-connected (FC) layer.
+Regarding the attack setting, three different backdoor attacks
+have been considered, as detailed below. For each setting,
+the training dataset is poisoned by considering 16 poisoning
+ratios α chosen in (0, 0.55]. For each α, 10 different poisoned
+training datasets are generated by choosing different classes
+as the target class.
+• Corrupted-label attack, with a 3×3 pixel trigger (abbrev.
+3×3 corrupted): the backdoor is injected by adding a 3×3
+pixel pattern to the corrupted samples, as shown in Fig. 2,
+and modifying the sample labels into that of the target class.
+• Corrupted-label attack, with ramp trigger (abbrev. ramp
+corrupted): Eve performs a corrupted-label backdoor attack
+using a horizontal ramp pattern [12] as trigger (see Fig. 2).
+The ramp pattern is defined as υ(i, j) = j∆/W, 1 ≤ i ≤ H,
+1 ≤ j ≤ W, where H × W is the size of the image and
+∆ is a parameter controlling the slope (and strength) of the
+ramp. We set ∆ = 40 in the experiments.
+• Clean-label attack, with 3×3 pixel trigger (abbrev. 3×3
+clean): the attack utilises the 3×3 pixel trigger pattern to
+perform a clean-label attack.
+2) Traffic signs: For the traffic sign classification task, we
+selected 16 different classes from the GTSRB dataset, namely,
+the most representative classes in the dataset, including 6
+speed-limit, 3 prohibition, 3 danger, and 4 mandatory signs.
+Each class has 1200 colour images with size 28 × 28. The
+model architecture used for training is based on ResNet18
+[43]. The feature representation is extracted from the 17-th
+layer, that is, before the FC layer, after an average pooling
+layer and ReLu activation. With regard to the attack, we
+considered the corrupted-label scenario. As triggering pattern,
+we considered a horizontal sinusoidal pattern, defined as
+υ(i, j) = ∆ sin(2πjf/W), 1 ≤ i ≤ H, 1 ≤ j ≤ W, where
+H × W is the size of input image. The parameters ∆ and f
+are used to control the strength and frequency of the trigger.
+In our experiment, we set ∆ = 20 and f = 6. As before, for a
+given α, the network is trained on 16 poisoned datasets, each
+time considering a different target classes. .
+3) Fashion clothes: Fashion-MNIST dataset includes 10
+classes of grey-level cloth images, each class consisting of
+6000 images of size 28×28. The model architecture used for
+the classification is based on AlexNet [44]. The representation
+used by the backdoor detector is extracted from the 5-th layer,
+at the output of the ReLu activation layer before the first FC
+layer. With regard to the attack, the poisoned samples are
+generated by performing the attack in a clean-label setting.
+A ramp trigger with ∆ = 256 is used to implement the
+attack. Once again, for each choice of α, the backdoor attack
+is repeated 10 times, each time considering a different target
+class.
+For all the classification tasks, the benign validation dataset
+Dval is obtained by randomly selecting 100 samples from all
+the classes in the dataset.
+C. Setting of defence parameters
+To implement the CCA-UD defence, we have to set the
+following parameters: the reduced dimension d′ for the clus-
+tering, the parameters of the DBSCAN algorithm, namely
+minPts and ϵ, and finally the threshold θ used by the
+clustering poisoning detection module. In our experiments, we
+set d′ = 2, minPts = 20 and ϵ = 0.8. This is the setting that,
+according to our experiments, achieves the best performance
+with the minimum complexity for the clustering algorithm
+(being d′ = 2). The effect of these parameters on the result of
+clustering and the detection performance is evaluated by the
+ablation study described in Section VI-A.
+With regard to θ, as mentioned before, AC, CI and CCA-
+UD involve the setting of a threshold for poisoning detection.
+For a fair comparison, we set the threshold in the same way
+for all the methods. In particular, we set θ by fixing the false
+positive rate. In general a value of θ results in different FPR
+rates for different classes. To avoid setting a different threshold
+for each class, then, we fixed it by setting the average FPR.
+In fact, setting the average FPR exactly may not be feasible,
+so we chose the threshold in such a way to minimize the
+distance from the target rate. Formally, by setting the target
+false positive rate to 0.05, the threshold θ∗ is determined as:
+θ∗ = arg min
+θ
+��0.05 − FPR(Dval)
+��.
+(10)
+VI. EXPERIMENTAL RESULTS
+In this section we report the results of the experiments we
+have carried out to evaluate the effectiveness of CCA-UD.
+A. Ablation study
+We start the experimental analysis with an ablation study
+investigating the effect of the three main hyperparameters of
+CCA-UD, namely d′ (regarding UMAP), and minPts and ϵ
+(for DBSCAN) on the effectiveness of the method. Based on
+this analysis, in all subsequent experiments we set d′ = 2,
+minPts = 20 and ϵ = 0.8.
+The influence of each parameter on the clustering result
+and the detection performance can be assessed by looking at
+Table I. The results refer to the case of MNIST classification,
+with backdoor poisoning performed by using a 3×3 pixel
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+9
+TABLE I: Ablation study on the three hyperparameters of CCA-UD. FPR and TPR for all cases are computed by letting
+θ = θ∗ as stated in Eq. (10). K and ζ are, respectively, the average number of clusters and the average fraction of outliers
+identified by DBSCAN.
+Hyperparameters
+BCB results
+BCP results
+PC results
+d′
+minP ts
+ϵ
+(K, ζ)
+F P R(BCB)
+(K, ζ)
+F P R(BCP )
+(K, ζ)
+T P R(P C)
+F P R(P C)
+AUC
+S1
+2
+20
+0.4
+(2.9, 0.005)
+0.050
+(4.3, 0.008)
+0.073
+(9.7, 0.003)
+1.000
+0.046
+0.998
+S2
+4
+20
+0.4
+(30.4, 0.097)
+0.044
+(22.6, 0.060)
+0.027
+(12.9, 0.012)
+0.432
+0.006
+0.989
+S3
+8
+20
+0.4
+(37.4, 0.142)
+0.066
+(23.7, 0.076)
+0.037
+(13.4, 0.012)
+0.448
+0.007
+0.990
+S4
+10
+20
+0.4
+(39.3, 0.153)
+0.057
+(24.5, 0.085)
+0.049
+(13.8, 0.013)
+0.501
+0.010
+0.987
+S5
+2
+3
+0.4
+(2.0, 0.000)
+0.050
+(2.2, 0.000)
+0.051
+(8.0, 0.000)
+1.000
+0.050
+1.000
+S6
+2
+10
+0.4
+(2.3, 0.001)
+0.050
+(2.6, 0.002)
+0.050
+(8.5, 0.001)
+1.000
+0.050
+0.999
+S7
+2
+20
+0.8
+(1.3, 0.000)
+0.050
+(1.6, 0.000)
+0.050
+(6.2, 0.000)
+1.000
+0.050
+1.000
+S8
+2
+20
+1.0
+(1.3, 0.000)
+0.049
+(1.6, 0.000)
+0.050
+(4.6, 0.000)
+1.000
+0.049
+1.000
+S9
+2
+20
+10.0
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+1.000
+1.000
+0.500
+S10
+10
+5
+0.4
+(15.5, 0.004)
+0.049
+(9.5, 0.002)
+0.068
+(11.9, 0.001)
+1.000
+0.046
+0.999
+S11
+10
+10
+0.4
+(17.8, 0.020)
+0.052
+(11.7, 0.012)
+0.077
+(10.6, 0.004)
+1.000
+0.030
+0.996
+S12
+10
+20
+0.2
+(29.2, 0.883)
+0.049
+(60.7, 0.732)
+0.045
+(111.3, 0.399)
+0.053
+0.031
+0.612
+S13
+10
+20
+0.6
+(2.0, 0.008)
+0.046
+(3.0, 0.004)
+0.042
+(7.6, 0.001)
+1.000
+0.042
+0.999
+S14
+10
+20
+1.0
+(1.2, 0.000)
+0.050
+(1.5, 0.000)
+0.050
+(6.2, 0.000)
+1.000
+0.049
+1.000
+S15
+10
+20
+3.0
+(1.1, 0.000)
+0.050
+(1.5, 0.000)
+0.050
+(3.9, 0.000)
+1.000
+0.050
+1.000
+S16
+10
+20
+10.0
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+1.000
+1.000
+0.500
+trigger pattern with label corruption. Similar considerations
+can be drawn in the other settings. The results in the table have
+been obtained by letting θ = θ⋆ as stated in Eq. (10). To start
+with, we observe that when utilising θ∗ in BCB and BCP
+cases, the FPR values is close to 0.05 for all the settings,
+while in the PC case FPR is close to or less than 0.05 for
+all settings except for S9 and S16, whes benign and poisoned
+samples collapse into a single cluster. In addition to TPR and
+FPR, the table shows the average number of clusters (K) and
+the average outlier ratio (ζ) identified by DBSCAN.
+From the first group of rows (S1-S4), we see that for a
+given setting of minPts and ϵ, increasing d′ leads to a larger
+average number of clusters and a larger fraction of outliers,
+as the DBSCAN algorithm results in a higher number of
+densely-connected regions. A similar behaviour is observed
+by increasing minPts or decreasing ϵ for a given d′ (second
+and third group of rows in the table). Expectedly, when ϵ
+is too large, e.g. 10, DBSCAN always results in one cluster
+thus failing to identify the poisoned samples. Based on the
+result in Table I, the settings S7 (d′ = 2, minPts = 20,
+ϵ = 0.8) and S15 (d′ = 10, minPts = 20, ϵ = 3) yield
+the best performance, the former having lower computational
+complexity, because of the lower dimension used to cluster
+the samples in the feature space (d′ = 2 instead of 10).
+B. Threshold setting
+The thresholds θ∗ obtained following the approach detailed
+in Section V-C for AC and CI and CCA-UD, are reported in
+Table II for the three different classification tasks considered
+in our experiments. Given that the threshold is set by relying
+on the validation dataset, it is necessary to verify that the target
+false positive rate (0.05 in our case) is also obtained on the
+test dataset. An excerpt of such results is shown in Table IV
+by referring to MNIST task (a similar behaviour is observed
+for the other classification tasks).
+Our experiments reveal that, for AC and CI, the threshold
+determined via Eq. (10) does not lead to a good operating
+point when used on the test dataset. In particular, while for
+CCA-UD, the threshold θ∗ set on the validation dataset yields
+a similar FPR (around 0.05) in the BCB, BCP and PC
+TABLE II: Values of θ∗ obtained for the various classification
+tasks.
+Method
+MNIST
+Traffic signs
+Fashion clothes
+AC
+0.335
+0.404
+0.301
+CI
+3.018
+1.673
+4.738
+CCA-UD
+0.950
+0.950
+0.950
+cases, this is not true for AC and CI, for which FPR(BCB),
+FPR(BCP ) and FPR(PC) are often smaller than 0.05,
+reaching 0 in many cases. This leads to a poor TPR(PC). In
+particular, with AC, when α > θ∗, both clusters are classified
+as benign, and then TPRα(PC) = FPRα(PC) = 0, even
+when the method would, in principle, be able to provide a
+perfect discrimination (AUCα ≈ 1). The difficulty in setting
+the threshold for AC and CI is also evident from the plots in
+Fig. 6, that report the FPR and TPR values averaged also
+on α, for different values of the threshold θ. From these plots,
+we immediately see that a threshold that works in all the cases
+can never be found for AC and CI.
+Due to the difficulties encountered to set the detection
+threshold for AC and CI10, the results at θ∗ for these methods
+are not reported in the other cases, that is, for traffic sign
+and fashion clothes classification, for which we report only
+the AUCα scores. Note that the possibility to set a unique
+threshold on a benign dataset that also works on poisoned
+datasets is very important for the practical applicability of a
+defence. Based on our results, CCA-UD has this remarkable
+property.
+C. Results on MNIST
+In this section, we evaluate the performance of CCA-UD
+against the three types of backdoor attacks, namely, 3×3
+corrupted, ramp corrupted, and 3×3 clean. Such performance
+as compared to those obtained by AC and CI. In Fig. 6, in each
+row, the three figures report the average performance of AC,
+CI and CCA-UD. The values of FPR(BCB), FPR(BCP ),
+TPR(PC) and FPR(PC) are reported for each method,
+as a function of the detection threshold θ. The behaviour of
+10Note that the problem of threshold setting is not addressed in the original
+papers, since different threshold are used in the various cases.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+10
+TABLE III: AUC scores of three methods in the three different
+attacks
+Method
+3×3 corrupted
+Ramp corrupted
+3×3 clean
+AC
+0.728
+0.733
+0.785
+CI
+0.964
+0.178
+0.488
+CCA-UD
+0.994
+0.996
+0.981
+FPR(Dval), which is utilised to determine the threshold θ∗
+(at 0.05 of FPR(Dval)), is also reported. The position of θ∗
+is indicated by a vertical dotted line.
+By observing the figure, we see that CCA-UD outperforms
+by far the other two methods in all the settings. In the first
+setting, we achieve TPR(PC) and FPR(PC) equal to 0.983
+and 0.051 at the optimal threshold θ∗, with FPR(BCB) =
+0.051 and FPR(BCP ) = 0.050. Instead, the performance
+achieved by AC and CI at their optimal threshold are very
+poor. Similar results are achieved for the second and third
+settings. In particular, for the second attack, CCA-UD achieves
+TPR(PC) and FPR(PC) equal to ( 0.975, 0.050) at θ∗, and
+(0.966, 0.050) for the third one.
+For a poisoned dataset, the AUC values obtained in the
+three settings are provided in Table III. From these results,
+we argue that CI has good discriminating capability (with
+an AUC only slightly lower than CCA-UD) against the first
+attack, but fails to defend against the other two. This is an
+expected behaviour since CI does not work when the triggering
+pattern is robust against average filtering, as it is the case of
+the ramp signal considered in the second attack, or with clean-
+label attacks, as it is the last setting.
+Table IV shows the results obtained for different values of
+the poisoning ratio α for the three different attacks. The values
+of FPR and TPR have been obtained by letting θ = θ∗.
+For the clean-label case, due to the difficulty of developing
+a successful attack [12], [27], [28], the backdoor can be
+successfully injected in the model only when α is large enough
+and, in any case, a successful attack could not always be
+obtained in the 10 repetitions. For this reason, in the third
+table, we report the number of successfully attacked classes
+(cnt) with different poisoning ratios. Upon inspection of Table
+IV, we observe that:
+• With regard to AC, the behaviour is similar under the three
+attack scenarios. Good results are achieved for intermediate
+values of α, namely in the [0.2, 0.3] range. When α < 0.134,
+instead, AUCα of AC is smaller than 0.786, and close
+to 0.5 for small α. In particular, AC cannot handle the
+backdoor attacks for which the poisoning ratio is smaller
+than 0.1. Moreover, when α > 0.5, AUCα goes to zero,
+as benign samples are judged as poisoned and vice-versa.
+Finally, by comparing the AUCα values in Fig. IVa and Fig.
+IVc, we see that AC achieves better performance against the
+corrupted-label attack than in the clean-label case.
+• With regard to CI, the detection performance achieved in
+the first attack scenario (3×3 corrupted) are good for all
+the values of α, with AUCα larger than 0.96 in most
+of the cases (with the exception of the smallest α, for
+which AUCα = 0.876), showing that CI can effectively
+defend against the backdoor attack in this setting, for every
+attack poisoning ratio. However, as expected, CI fails in the
+other settings, with AUCα lower than 0.5 in all the cases,
+confirming the limitations mentioned in Section III-A2.
+• Regarding CCA-UD, good results are achieved in all the-
+cases and for every value of α, with a perfect or nearly
+perfect AUCαin most of the cases. Moreover, by letting
+θ = θ∗, a very good TPRα(PC) is obtained, larger
+than 0.95 in almost all the cases, with FPRα(BCP ) and
+FPRα(PC) around 0.05. Overall, the tables prove the
+universality of CCA-UD that works very well regardless of
+the specific attack setting and regardless of the value of α.
+Note, since CCA-UD achieves a larger AUCα than AC and
+CI, CCA-UD outperforms AC and CI not only when θ = θ∗
+but also when θ is set adaptively.
+Finally, these results show that CCA-UD can effectively
+defend against both corrupted and clean-label attacks, thus
+confirming that the strategy used to detect poisoned clusters
+exploits a general misclassification behaviour present in both
+corrupted- and clean-label attacks.
+D. Results on Traffic Signs
+Fig. 7a-7c show the average performance of AC, CI, and
+CCA-UD on the traffic signs task. Similar considerations
+to the MNIST case can be made. CCA-UD achieves very
+good average performance at the operating point given by θ∗,
+where TPR(PC) and FPR(PC) are ( 0.965, 0.058) (with
+FPR(BCB) = FPR(BCB) ≈ 0.08), while for AC and CI
+a threshold that works well on the average can not be found.
+In the case of a poisoned dataset, the average AUC of the
+detection AUC is equal to 0.897, 0.958, 0.993 for AC, CI,
+and CCA-UD, respectively.
+We observe that CI gets a good AUC, too. In fact, in
+this case, given that the size of the input image is 28×28,
+the triggering pattern, namely the sinusoidal signal can be
+effectively removed by a 5 × 5 average filter.
+The results obtained for various α are reported in Table Va.
+As it can be seen, CCA-UD gets very good performance in
+terms of TPRα(PC) and FPRα(PC) measured at θ = θ∗
+in all the cases. The AUCα is also larger than that achieved
+by AC and CI for all values of α. As observed before, while
+CI is relatively insensitive to α, the performance of AC drop
+when α < 0.1 or α > 0.5.
+E. Results on Fashion Clothes
+Fig. 7d-7f report the results obtained by AC, CI, and CCA-
+UD on the fashion clothes task. Once again, the performance
+achieved by CCA-UD are largely superior to those achieved by
+AC and CI. In particular, by looking at Fig. 7d-7f, CCA-UD
+achieves TPR(PC) and FPR(PC) equal to (1.000, 0.053),
+with FPR(BCB) = FPR(BCP ) ≈ 0.05. Regarding the
+AUC scores, AUC of AC, CI, and CCA-UD are 0.900, 0.106,
+0.997 respectively. Since the attack is carried out in a clean-
+label modality, the poor performance of CI were expected. The
+results for various α, reported in Table Vb, confirm the same
+behaviour, with CCA-UD getting very good performance in
+all the cases, always overcoming the other two methods.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+11
+(a) AC in 3×3 corrupted
+(b) CI in 3×3 corrupted
+(c) CCA-UD in 3×3 corrupted
+(d) AC in ramp corrupted
+(e) CI in ramp corrupted
+(f) CCA-UD in ramp corrupted
+(g) AC in 3×3 clean
+(h) CI in 3×3 clean
+(i) CCA-UD in 3×3 clean
+Fig. 6: Average performance of AC and CI, and CCA-UD for different values of the threshold against the three types of
+backdoor attacks implemented in the case of MNIST classification. From top to bottom the plots refer to 3×3 corrupted in
+(a)-(c), ramp corrupted in (d)-(f), and 3×3 clean in (g)-(i). From left to right we report the performance of AC, CI and
+CCA-UD. The position of θ∗ is indicated by a vertical dotted line.
+TABLE IV: Performance of AC, CI and CCA-UD for various poisoning ratios α, against the three types of backdoor attacks
+for MNIST classification, The FPR and TPR values are computed at θ = θ∗. In the 3 × 3 table cnt indicates the number of
+successful attacks in 10 repetitions.
+AC
+CI
+CCA-UD
+α
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+0.025
+0.025
+0.000
+0.000
+0.563
+0.012
+0.324
+0.022
+0.876
+0.050
+0.908
+0.051
+0.949
+0.050
+0.055
+0.099
+0.000
+0.628
+0.005
+0.581
+0.001
+0.977
+0.050
+0.989
+0.050
+0.994
+0.096
+0.000
+0.395
+0.000
+0.757
+0.005
+0.654
+0.000
+0.996
+0.050
+0.999
+0.050
+0.999
+0.134
+0.000
+0.792
+0.000
+0.958
+0.009
+0.559
+0.002
+0.990
+0.051
+0.999
+0.050
+1.000
+0.186
+0.000
+0.994
+0.000
+0.997
+0.000
+0.577
+0.001
+0.985
+0.050
+1.000
+0.050
+1.000
+0.258
+0.000
+0.993
+0.000
+0.997
+0.014
+0.540
+0.070
+0.961
+0.050
+1.000
+0.050
+1.000
+0.359
+0.000
+0.000
+0.000
+0.998
+0.000
+0.571
+0.005
+0.964
+0.050
+1.000
+0.050
+1.000
+0.550
+0.000
+0.000
+0.000
+0.001
+0.000
+0.829
+0.000
+0.953
+0.050
+1.000
+0.050
+1.000
+(a) 3×3 corrupted
+AC
+CI
+CCA-UD
+α
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+0.035
+0.000
+0.050
+0.024
+0.593
+0.009
+0.000
+0.008
+0.407
+0.051
+0.871
+0.050
+0.966
+0.050
+0.024
+0.090
+0.028
+0.593
+0.000
+0.000
+0.000
+0.119
+0.050
+0.914
+0.050
+0.998
+0.096
+0.000
+0.400
+0.000
+0.786
+0.003
+0.000
+0.000
+0.216
+0.050
+0.989
+0.050
+0.998
+0.134
+0.024
+0.798
+0.001
+0.962
+0.019
+0.000
+0.000
+0.142
+0.050
+0.999
+0.050
+0.998
+0.186
+0.000
+0.992
+0.003
+0.995
+0.107
+0.000
+0.000
+0.179
+0.051
+1.000
+0.050
+1.000
+0.258
+0.025
+0.999
+0.000
+0.999
+0.000
+0.000
+0.000
+0.088
+0.050
+1.000
+0.050
+1.000
+0.359
+0.025
+0.000
+0.000
+0.999
+0.021
+0.000
+0.000
+0.144
+0.051
+1.000
+0.050
+1.000
+0.550
+0.000
+0.000
+0.000
+0.002
+0.004
+0.000
+0.000
+0.135
+0.050
+1.000
+0.050
+1.000
+(b) Ramp corrupted
+AC
+CI
+CCA-UD
+α
+cnt
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+0.050
+2
+0.000
+0.000
+0.000
+0.441
+0.000
+0.683
+0.835
+0.438
+0.051
+0.642
+0.050
+0.809
+0.069
+3
+0.000
+0.000
+0.000
+0.533
+0.000
+0.667
+0.667
+0.296
+0.050
+0.952
+0.050
+0.972
+0.096
+3
+0.000
+0.000
+0.000
+0.528
+0.000
+0.333
+0.333
+0.595
+0.050
+0.951
+0.050
+0.972
+0.134
+3
+0.000
+0.000
+0.000
+0.610
+0.000
+0.667
+0.667
+0.539
+0.050
+0.975
+0.050
+0.987
+0.186
+5
+0.000
+0.384
+0.003
+0.746
+0.000
+0.600
+0.600
+0.471
+0.051
+0.982
+0.050
+0.991
+0.258
+5
+0.000
+0.929
+0.011
+0.959
+0.000
+0.601
+0.644
+0.516
+0.050
+0.994
+0.051
+0.996
+0.359
+5
+0.000
+0.315
+0.000
+0.975
+0.000
+0.206
+0.213
+0.437
+0.050
+0.993
+0.050
+0.996
+0.450
+5
+0.000
+0.000
+0.000
+0.969
+0.009
+0.729
+0.786
+0.554
+0.050
+0.997
+0.050
+0.998
+(c) 3×3 clean
+
+1.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.5
+0.1
+0.3
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+pel
+0.2
+0.0
+1
+2
+3
+5
+4
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+age
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+age
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+ge
+FPR(BCB)
+centa
+0.6
+FPR(BCp)
+g
+0.4
+FPR(D)
+pel
+0.2
+0.0
+0
+2
+3
+4
+5
+L
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+ercentage
+FPR(BCB)
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.6
+0.8
+0.0
+0.2
+0.4
+1.0
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.1
+0.2
+0.3
+0.5
+0.4
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+FPR(BCB)
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+2
+3
+L
+4
+51.0
+TPR(PC)
+0.8
+FPR(PC)
+ge
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+12
+(a) AC in traffic signs task
+(b) CI in traffic signs task
+(c) CCA-UD in traffic signs task
+(d) AC in fashion clothes task
+(e) CI in fashion clothes task
+(f) CCA-UD in fashion clothes task
+Fig. 7: Average performance of AC, CI, and CCA-UD for different values of θ for the traffic signs and fashion clothes task.
+The vertical dotted line indicates the position of θ∗ for the various methods.
+TABLE V: Performance of AC, CI, and CCA-UD for various
+poisoning ratios for the traffic sign and fashion cloth task. The
+FPR and TPR values are computed at θ = θ∗. Since for AC
+and CI it is not possible to find a unique value of θ working
+in all conditions, we report only the AUC values.
+AC
+CI
+CCA-UD
+α
+cnt
+AUCα
+AUCα
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+0.050
+9
+0.793
+0.923
+0.983
+0.073
+0.946
+0.061
+0.096
+9
+0.850
+0.928
+0.991
+0.058
+0.998
+0.059
+0.134
+9
+0.949
+0.959
+0.992
+0.057
+0.998
+0.057
+0.186
+10
+0.958
+0.965
+0.993
+0.064
+0.999
+0.056
+0.359
+13
+0.946
+0.965
+0.996
+0.086
+0.985
+0.054
+0.450
+14
+0.917
+0.965
+0.994
+0.070
+0.980
+0.055
+0.550
+15
+0.869
+0.996
+0.999
+0.059
+0.999
+0.051
+(a) Traffic signs
+AC
+CI
+CCA-UD
+α
+cnt
+AUCα
+AUCα
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+0.069
+3
+0.618
+0.056
+0.998
+0.053
+1.000
+0.052
+0.096
+3
+0.513
+0.341
+0.995
+0.054
+1.000
+0.056
+0.134
+3
+0.940
+0.087
+0.998
+0.059
+1.000
+0.053
+0.186
+4
+1.000
+0.037
+0.998
+0.054
+1.000
+0.055
+0.258
+5
+1.000
+0.083
+0.996
+0.055
+1.000
+0.057
+0.359
+5
+1.000
+0.015
+0.998
+0.056
+1.000
+0.052
+0.450
+5
+1.000
+0.174
+1.000
+0.055
+1.000
+0.050
+(b) Fashion clothes
+VII. CONCLUDING REMARKS
+We have proposed a universal backdoor detection method,
+called CCA-UD, aiming at revealing the possible presence of a
+backdoor inside a model and identify the poisoned samples by
+analysing the training dataset. CCA-UD relies on DBSCAN
+clustering and on a new strategy for the detection of poisoned
+clusters based on the computation of clusters’ centroids. The
+capability of the centroids’ features to cause a misclassification
+of benign samples is exploited to decide whether a cluster is
+poisoned or not. We evaluated the effectiveness of CCA-UD
+on a wide variety of classification tasks and attack scenarios.
+The results confirm that the method can work regardless of the
+corruption strategy (corrupted and clean label setting) and the
+type of trigger used by the attacker (local or global pattern).
+Moreover, the method is effective regardless of the poisoning
+ratio used by the attacker, that can be either very small or even
+larger than 0.5. Furthermore, we proved that the performance
+achieved by CCA-UD are always superior to those achieved
+by the existing methods, also when these methods are applied
+in a scenario that meets their operating requirements.
+Future work will be devoted to the analysis of the behaviour
+of the proposed method against multiple triggers attacks, that
+is when multiple triggers are used to poison the samples,
+possibly to induce more than one malicious behaviour inside
+the network. The capability of the method to defend against
+backdoor attacks in application scenarios beyond image clas-
+sification, is also worth investigation.
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+Solving Unsplittable Network Flow Problems with
+Decision Diagrams
+Hosseinali Salemi, Danial Davarnia
+Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, IA 50011,
+[email protected], [email protected]
+In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the
+incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called no-split
+no-merge requirement arises in unit train scheduling where train consists should remain intact at stations
+that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network
+flow problems with standard mixed-integer programming formulations is computationally difficult due to
+the large number of binary variables needed to determine matching pairs between incoming and outgoing
+arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit
+train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based
+framework that decomposes the underlying two-stage formulation into a master problem that contains the
+combinatorial requirements, and a subproblem that models a continuous network flow problem. The master
+problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a
+substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems
+output cutting planes that are used to refine the master DD. Computational experiments show a significant
+improvement in solution time of the DD framework compared with that of standard methods.
+Key words : Decision Diagrams; Network Optimization; Mixed Integer Programs; Unit Trains;
+Transportation
+History :
+1.
+Introduction
+Over the past several decades, rail freight transportation has continued to grow as the prime
+means of transportation for high-volume commodities. Advantages of rail transportation include
+reliability, safety, cost-efficiency and environmental-sustainability as compared with alternative
+methods of transportation. In terms of scale, the rail network accounted for 27.2 percent of U.S.
+freight shipment by ton-miles in 2018 (Furchtgott-Roth et al. 2021); see Figure 1. The Federal
+Highway Administration estimates that the total U.S. freight shipments will be 24.1 billion tons
+in 2040, a 30 percent increase from the 2018 total transportation of 18.6 billion tons. With the
+purpose of meeting such market growth, America’s freight railway companies have invested nearly
+1
+arXiv:2301.01844v1  [math.OC]  4 Jan 2023
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+2
+$740 billion on capital expenditures and maintenance from 1980 to 2020 (Association of American
+Railroads 2021).
+Figure 1
+Pie chart for ton-miles of freight shipments by mode within the U.S. in 2018. Multiple modes includes
+mail. Air and truck-air with the share of 0.1% are omitted.
+To reduce rail freight transportation costs and shipment delays, railroad companies offer unit
+train services for carrying high-volume products. Unit trains haul a single type freight in a way that
+no car is attached or detached while the cargo train is on its way from an origin to a destination,
+except in specific locations that are equipped with required manpower and machinery. These trains
+usually operate all day, use dedicated equipment, and can be loaded/unloaded in 24 hours. They
+are known to be one of the fastest and most efficient means of railroad transportation. (Association
+of American Railroads 2021). Traditionally, unit trains are used to carry bulk cargo such as coal,
+grain, cement, and rock. Bulk liquids like crude oil and food such as wheat and corn are also
+shipped by unit trains. According to the Federal Railroad Administration data, bulk commodities
+account for 91 percent of the U.S. railroad freights. Approximately all coal shipped through railways
+in the U.S. are transported by unit trains. Moreover, these trains contribute significantly to the
+shipping process of crude oil as each unit train is capable of carrying 85,000 barrels (Association
+of American Railroads 2021). In an operational level, the core unit train model can be described
+as follows. Given a set of supply, intermediate, and demand locations in a railroad network, the
+unit train scheduling problem seeks to find optimal routes for unit trains to send flows from supply
+to demand points with the objective of minimizing the total transportation cost while meeting
+demand of customers, respecting capacities of tracks, and satisfying no-car attaching/detaching
+requirements in specific locations. As a result, designing blocking plans to determine locations
+
+Multiple modes
+8%
+Pipeline, 19%
+Truck, 39%
+Water, 7%
+Rail, 27%Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+3
+where cars need to be switched between trains is irrelevant in this problem, unlike scheduling other
+types of trains (Davarnia et al. 2019).
+Despite the significance of unit train scheduling, exact optimization approaches to solve associ-
+ated problems are scarce, partially due to their structural complexities. One of the main challenges
+in modeling unit trains is the requirement that the train consists must remain intact when passing
+through stations that lack necessary busting/formation equipment. In optimization, this require-
+ment is referred to as no-split no-merge (NSNM), which guarantees that the flows entering to or
+exiting from certain nodes of the unit train network cannot be split or merged. Incorporating this
+requirement into typical transportation network models yields the so-called generalized unsplittable
+flow problem (GUFP), where the objective is to determine the minimum-cost unit train sched-
+ules that satisfy the given demand. Numerous studies have shown that considering deterministic
+demands might result in the complete failure of the transportation scheduling (Demir et al. 2016,
+Layeb et al. 2018), motivating the study of stochastic variants of the unit train scheduling problems
+where the demand is uncertain. As a result, in this paper, we consider a stochastic variant of the
+GUFP, referred to SGUFP, that is modeled as a two-stage optimization problem. The first stage
+decides a matching between the incoming and outgoing arcs of the nodes of the railroad network,
+and the second stage determines the amount of flow that should be sent through the matching arcs
+of the network to satisfy the uncertain demand represented by a number of demand scenarios. We
+propose a novel exact solution framework to solve this problem in the operational level.
+Our proposed methodology is based on decision diagrams (DDs), which are compact graphical
+data structures. DDs were initially introduced to represent boolean functions with applications in
+circuit design. Over the past decade, researchers have successfully extended DDs domain by devel-
+oping DD-based algorithms to solve discrete optimization problems in different areas of application.
+Because of its structural limitation to model integer programs only, DDs have never been used
+to solve transportation problems that inherently include continuous variables. In this paper, we
+extend the application scope of DDs by introducing a novel framework that is capable of modeling
+network problems with both integer and continuous components as in the SGUFP.
+1.1.
+Literature Review on Train Scheduling
+Many variants of train routing and scheduling problems with different objective functions and
+set of constraints under deterministic and stochastic conditions have been introduced and vastly
+studied in the literature; see surveys by Cordeau, Toth, and Vigo (1998), Harrod and Gorman
+(2010), Lusby et al. (2011), Cacchiani and Toth (2012), and Turner et al. (2016) for different
+problems classifications and structures. Mixed integer linear and nonlinear programming formu-
+lations are among the most frequent exact approaches to model different classes of these prob-
+lems (Jovanovi´c and Harker 1991, Huntley et al. 1995, Sherali and Suharko 1998, Lawley et al.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+4
+2008, Haahr and Lusby 2017, Davarnia et al. 2019). Proposed solution techniques include but are
+not limited to branch-and-bound methods (Jovanovi´c and Harker 1991, Fuchsberger and L¨uthi
+2007), branch-and-cut frameworks (Zwaneveld, Kroon, and Van Hoesel 2001, Ceselli et al. 2008),
+branch-and-price approaches (Lusby 2008, Lin and Kwan 2016), graph coloring algorithms (Cor-
+nelsen and Di Stefano 2007), and heuristics (Carey and Crawford 2007, Liu and Kozan 2011, I¸cy¨uz
+et al. 2016). Rolling stock scheduling (Abbink et al. 2004, Alfieri et al. 2006, Haahr et al. 2016,
+Bornd¨orfer et al. 2016) that assigns rolling stocks to a given timetable, and crew scheduling (Kwan
+2011, Shen et al. 2013, Heil, Hoffmann, and Buscher 2020) that covers train activities by assigning
+crews to the associated operations are other major problems arising in the area of railroad planning.
+Due to the inherent uncertainty in different types of train scheduling and routing problems, many
+researchers have studied stochastic variants of the problems where the supply/demand is considered
+to be uncertain. Jordan and Turnquist (1983) propose a model for railroad car distribution where
+supply and demand of cars are uncertain. Jin et al. (2019) study a chance-constrained programming
+model for the train stop planning problem under stochastic demand. Ying, Chow, and Chin (2020)
+propose a deep reinforcement learning approach for train scheduling where the passenger demand
+is uncertain. Recently, Gong et al. (2021) propose a stochastic optimization method to solve a train
+timetabling problem with uncertain passenger demand. Also see works by Meng and Zhou (2011),
+Quaglietta, Corman, and Goverde (2013), Larsen et al. (2014) that consider train dispatching
+problems under stochastic environments.
+In the context of unit train scheduling, Lawley et al. (2008) study a time-space network flow
+model to schedule bulk railroad deliveries for unit trains. In their model, the authors consider char-
+acteristics of underlying rail network, demands of customers, and capacities of tracks, stations, and
+loading/unloading requirements. They propose a mixed integer programming (MIP) formulation
+that maximizes the demand satisfaction while minimizing the waiting time at stations. Lin and
+Kwan (2014) (cf. Lin and Kwan (2016)) propose a model for a train scheduling problem that is capa-
+ble to capture locations where coupling/decoupling is forbidden. They develop a branch-and-price
+algorithm inspired by column generation to solve the associated problem. Lin and Kwan (2018)
+also propose a heuristic branch-and-bound approach to decrease coupling/decoupling redundancy.
+I¸cy¨uz et al. (2016) study the problem of planning coal unit trains that includes train formation,
+routing, and scheduling. As noted by the authors, their proposed MIP formulation fails to solve
+the problem directly due to its large size. As a remedy, they develop a time-efficient heuristic that
+produces good quality solutions. More recently, Davarnia et al. (2019) introduce and study the
+GUFP with application to unit train scheduling. In particular, the authors show how to impose
+NSNM restrictions in network optimization problems. They present a polyhedral study and pro-
+pose a MIP formulation to model a stylized variant of the unit train scheduling problem. In the
+present paper, we use their formulation (see section 3.1) as a basis for our solution framework.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+5
+The unsplittable flow problem (UFP) was first introduced by Kleinberg (1996) as a generalization
+of the disjoint path problem. Given a network with capacities for arcs and a set of source-terminal
+vertex pairs with associated demands and rewards, the objective in the UFP is to maximize the
+total revenue by selecting a subset of source-terminal pairs and routing flows through a single
+path for each of them to satisfy the associated demand. In the GUFP, however, there can exist
+nodes that do not need to respect the NSNM requirement, and demands can be satisfied by
+passing flows through multiple paths. It is well-known that different variants of UFP are NP-
+hard (Baier, K¨ohler, and Skutella 2005, Kolman and Scheideler 2006, Chakrabarti et al. 2007). Since
+its introduction, the UFP structure has been used in different areas of application, from bandwidth
+allocation in heterogeneous networks (Kolman and Scheideler 2006), to survivable connection-
+oriented networks (Walkowiak 2006), and virtual circuit routing problems (Hu, Lan, and Wan
+2009). Considering the hardness of the problem, approximation algorithms have been a common
+technique to tackle different variants of the UFP in the literature (Baier, K¨ohler, and Skutella
+2005, Chakrabarti et al. 2007).
+1.2.
+Literature Review on Decision Diagrams
+DDs are directed acyclic graphs with a source and a terminal node where each source-terminal path
+encodes a feasible solution to an optimization problem. In DDs, each layer from the source to the
+terminal represents a decision variable where labels of arcs show their values. Had˘zi´c and Hooker
+(2006) proposed to use DDs to model the feasible region of a discrete optimization problem and used
+it for postoptimality analysis. Later, Andersen et al. (2007) presented relaxed DDs to circumvent
+the exponential growth rate in the DD size when modeling large discrete optimization problems.
+Bergman et al. (2016b) introduced a branch-and-bound algorithm that iteratively uses relaxed and
+restricted DDs to find optimal solution. The literature contains many successful utilization of DDs
+in different domains; see works by Bergman and Cire (2018), Serra and Hooker (2019), Davarnia
+and Van Hoeve (2020), Gonzalez et al. (2020), and Hosseininasab and Van Hoeve (2021) for some
+examples.
+Until recently, applications of DDs were limited to discrete problems, and the question on how to
+use DDs in solving optimization problems with continuous variables was unanswered. To address
+this limitation, Davarnia (2021) proposed a technique called arc-reduction that generates a DD
+that represents a relaxation of the underlying continuous problem. In a follow-up work, Salemi
+and Davarnia (2022a) established necessary and sufficient conditions for a general MIP to be
+representable by DDs. They showed that a bounded MIP can be remodeled and solved with DDs
+through employing a specialized Benders decomposition technique. In this paper, we build on this
+framework to design a novel DD-based methodology to solve the SGUFP.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+6
+1.3.
+Contributions
+While there are several studies in the literature dedicated to the unit train problem, exact method-
+ologies that provide a rigorous treatment of the NSNM requirement at the heart of unit train
+models are scarce. In this paper, we design a novel exact DD-based framework to solve the SGUFP,
+as a more realistic and more challenging variant of this problem class. To our knowledge, this is
+the first work that studies SGUFP from an exact perspective, and the first application of DDs
+to a transportation problem. Our proposed framework formulates the problem in a transformed
+space of variables, which has a smaller dimension compared to the standard MIP formulations
+of the SGUFP. This presentation mitigates the computational difficulties stemmed from the MIP
+formulation size, providing a viable solution approach for large-scale network problems. The core
+principles of our DD framework can also be used to model other transportation problems with
+similar structure, as an alternative to traditional network optimization techniques.
+The remainder of this paper is organized as follows. In Section 2 we provide basic definitions
+and a brief overview on discrete and continuous DD models, including the DD-BD method to
+solve bounded MIPs. In Section 3, we adapt the DD-BD method to solve the SGUFP. We propose
+algorithms to construct exact and relaxed DDs to solve the problem in a transformed space.
+Section 4 presents computational experiments to evaluate the performance of the DD-BD method
+for the SGUFP. We give concluding remarks in Section 5.
+2.
+Background on DDs
+In this section, we present basic definitions and results relevant to our DD analysis.
+2.1.
+Overview
+A DD D = (U,A,l) with node set U, arc set A, and arc label mapping l : A → R is a directed acyclic
+graph with n ∈ N arc layers A1,A2,...,An, and n + 1 node layers U1,U2,...,Un+1. The node layers
+U1 and Un+1, with |U1| = |Un+1| = 1, contain the root r and the terminal t, respectively. In any arc
+layer j ∈ [n] := {1,2,...,n}, an arc (u,v) ∈ Aj is directed from the tail node u ∈ Uj to the head node
+v ∈ Uj+1. The width of D is defined as the size of its largest Uj. DDs can model a bounded integer
+set P ⊆ Zn in such a way that each r-t arc-sequence (path) of the form (a1,...,an) ∈ A1 × ... × An
+encodes a point y ∈ P where l(aj) = yj for j ∈ [n], that is y is an n-dimensional point in P whose
+j-th coordinate is equal to the label value l(aj) of arc the aj. For such a DD, we have P = Sol(D),
+where Sol(D) denotes the finite collection of all r-t paths.
+The graphical property of DDs can be exploited to optimize an objective function over a discrete
+set P. To this end, DD arcs are weighted in such a way that the cumulative weight of an r-t
+path that encodes a solution y ∈ P equals to the objective function value evaluated at y. Then, a
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+7
+shortest (resp. longest) r-t path for the underlying minimization (resp. maximization) problem is
+found, an operation that can be performed in polynomial time.
+The construction of an exact DD as described above is computationally prohibitive due to the
+exponential growth rate of its size. To alleviate this difficulty, relaxed and restricted DDs are
+proposed to keep the size of DDs under control. In a relaxed DD, nodes are merged in such a way
+that the width of the resulting diagram is bounded by a predetermined width limit. This node-
+merging process ensures that all feasible solutions of the original set are encoded by a subset of
+all r-t paths in the resulting DD. Optimization over this relaxed DD provides a dual bound to the
+optimal solution of the original problems. In a restricted DD, the collection of all r-t paths of the
+DD encode a subset of the feasible solutions of the original set. Optimization over this restricted
+DD provides a primal bound to the optimal solution of the original problems. The restricted and
+relaxed DDs can be iteratively refined in a branch-and-bound scheme to find the optimal value of
+a problem through convergence of their primal and dual bounds. The following example illustrates
+an exact, relaxed and restricted DD for a discrete optimization problem.
+Example 1. Consider the discrete optimization problem max{5y1 + 10y2 + 4y3 | y ∈ P} where
+P = {(1,0,0),(1,0,1),(0,1,0),(0,0,1),(0,0,0)}. The exact DD D with width 3 in Figure 2(a) models
+the feasible region P. The weight of each arc a ∈ Aj, for j ∈ {1,2,3}, shows the contribution of
+variable yj’s value assignment to the objective function. The longest r-t path that encodes the
+optimal solution (y∗
+1,y∗
+2,y∗
+3) = (0,1,0) has length 10, which is the optimal value to the problem.
+By reducing the width limit to 2, we can build relaxed and restricted DDs for P as follows. The
+relaxed DD D in Figure 2(b) provides an upper bound to the optimal solution, where the longest
+path with length 14 is obtained by an infeasible point (y1,y2,y3) = (0,1,1). Finally, the restricted
+DD D in Figure 2(c) gives a lower bound to the optimal solution, where the longest path with
+length 9 encodes a feasible solution (y1,y2,y3) = (1,0,1).
+2.2.
+Continuous DD Models
+While the framework described in the previous section can be applied to solve different classes of
+discrete optimization problems, its extension to model sets with continuous variables requires a
+fundamentally different approach. The reason that the traditional DD structure is not viable for
+continuous sets is that representing the domain of a continuous variable through arcs requires an
+infinite number of them, spanning all values within a continuous interval, which is structurally
+prohibitive in DD graphs. Fortunately, there is a way to overcome this obstacle by decomposing
+the underlying set into certain rectangular formations, which can in turn be represented through
+node-sequences in DDs. In what follows, we give an overview of these results as relevant to our
+analysis.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+8
+r
+t
+5
+0
+0
+10
+0
+0
+4
+0
+4
+0
+(a) Exact DD D
+r
+t
+5
+0
+0
+10
+0
+0
+4
+4
+0
+(b) Relaxed DD D
+r
+t
+5
+0
+0
+0
+0
+4
+4
+0
+y1
+y2
+y3
+(c) Restricted DD D
+Figure 2
+The exact, relaxed, and restricted DDs representing P in Example 1. Solid and dotted arcs indicate
+one and zero arc labels, respectively. Numbers next to arcs represent weights.
+Consider a bounded set P ⊆ Rn. Salemi and Davarnia (2022a) give necessary and sufficient
+conditions for P to admit the desired rectangular decomposition. Such a set is said to be DD-
+representable w.r.t. a fixed index set I ⊆ [n], as there exists a DD D such that max{f(x) | x ∈
+P} = max{f(x) | x ∈ Sol(D)} for every function f(x) that is convex in the space of variables xI.
+A special case of DD-representable sets is given next.
+Proposition 1. Any bounded mixed integer set of the form P ⊆ Zn × R is DD-representable
+w.r.t. I = {n + 1}.
+□
+This result gives rise to a novel DD-based framework to solve general bounded MIPs as outlined
+below. Consider a bounded MIP H := max{cy +dx | Ay +Gx ≤ b, y ∈ Zn}. Using Benders decom-
+position (BD), formulation H is equivalent to maxy∈Zn{cy +maxx{dx | Gx ≤ b−Ay}}, which can
+be reformulated as M = max{cy +z | (y;z) ∈ Zn ×[l,u]}, where l,u ∈ R are some valid bounds on z
+induced from the boundedness of H. Here, M is the master problem and z represents the objective
+value of the subproblem maxx{dx | Gx ≤ b − A¯y} for any given ¯y as an optimal solution of the
+master problem. The outcome of the subproblems is either an optimality cut or a feasibility cut
+that will be added to the master problem. Then, the master problem will be resolved. Proposition 1
+implies that formulation M can be directly modeled and solved with DDs. For this DD, we assign n
+arc layers to the integer variables y1,y2,...,yn, and one arc layer to the continuous variable z with
+only two arc labels showing a lower and upper bound for this variable. To find an optimal solution,
+the longest path is calculated, which will be used to solve the subproblems. Note that since M is
+a maximization problem, a longest path of the associated DD encodes an optimal solution, and
+its length gives the optimal value; see Example 2. The feasibility and optimality cuts generated
+by the subproblems will then be added to refine the DD, whose longest path will be recalculated.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+9
+The refinement technique consists of removing arcs of the DD that lead to solutions that violate
+the added inequality, as well as splitting nodes of the DD that lead to different subsequent partial
+assignments; see Bergman et al. (2016a) for a detailed account on DD refinement techniques. We
+illustrate this approach in Example 2.
+Example 2. Suppose that max{2y1 + 4y2 + z | y ∈ P,z ≤ 25} forms the master problem at the
+penultimate iteration of a BD algorithm, where P = {(0,0),(1,1)}. This problem is represented
+by the DD D in Figure 3(a) where −M is a valid lower bound for z. The longest path of D
+encodes the solution (ˆy1, ˆy2, ˆz) = (1,1,25). Assume that using the point (ˆy1, ˆy2) = (1,1) in the
+associated subproblem generates an optimality cut z ≤ 3y1 + 2y2 + 10 for the final iteration of the
+BD algorithm. Refining DD D with respect to this cut yields the new DD in Figure 3(b). The
+longest path represents the optimal solution (y∗
+1,y∗
+2,z∗) = (1,1,15) with length 21, which is the
+optimal value.
+r
+t
+2
+0
+4
+0
+−M
+25
+25
+−M
+(a) penultimate iteration
+r
+t
+2
+0
+4
+0
+−M
+15
+10
+−M
+y1
+y2
+z
+(b) final iteration
+Figure 3
+The last two iterations of solving the master problem in Example 2
+Using the DD framework as outlined above can be computationally challenging due to exponen-
+tial growth rate of the size of an exact DD. To mitigate this difficulty, restricted/relaxed DDs can
+be employed inside of the BD framework as demonstrated in Algorithm 1. We refer to this solution
+method as DD-BD (Salemi and Davarnia 2022a).
+In explaining the steps of Algorithm 1, let point ˆy ∈ Zk, where k ≤ n, be a partial value assignment
+to the first k coordinates of variable y, i.e., yi = ˆyi for all i ∈ [k]. We record the set of all partial
+value assignments in ˆY = {ˆy ∈ Zk | k ∈ [n]}∪{⊖}, where ⊖ represents the case where no coordinate
+of y is fixed. Set C contains the produced Benders cuts throughout the algorithm, and we denote
+the feasible region described by these cuts by F C. Further, define MC(ˆy) = max{cy + z | (y;z) ∈
+Zn × [l,u] ∩ F C, yi = ˆyi,∀i ∈ [k]} to be the restricted master problem M obtained through adding
+cuts in C and fixing the partial assignment ˆy. In this definition, the case with C = ∅ and ˆY = {⊖}
+is denoted by M∅(⊖) = M, which is an input to Algorithm 1.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+10
+Algorithm 1: DD-BD
+Data: MIP H, construction method to build restricted and relaxed DDs for M
+Result: An optimal solution (y∗,z∗) and optimal value w∗ to H
+1 initialize set of partial assignments ˆY = {⊖}, set of Benders cuts C = ∅, and w∗ = −∞
+2 if ˆY = ∅ then
+3
+terminate and return (y∗,z∗) and w∗
+4 else
+5
+select ˆy ∈ ˆY and update ˆY ← ˆY \ {ˆy}
+6
+create a restricted DD D associated with MC(ˆy)
+7
+if D ̸= ∅ then
+8
+find a longest r-t path of D with encoding point (y,z) and length w
+9
+solve the BD subproblem using y to obtain Benders cut C
+10
+if C ∈ C then
+11
+go to line 17
+12
+else
+13
+update C ← C ∪ C and refine D w.r.t. C
+14
+go to line 8
+15
+else
+16
+go to line 2
+17
+if w > w∗ then
+18
+update w∗ ← w and (y∗,z∗) ← (y,z)
+19
+if D provides an exact representation of MC(ˆy) then
+20
+go to line 2
+21
+else
+22
+create a relaxed DD D associated with MC(ˆy)
+23
+find a longest r-t path of D with length w
+24
+if w > w∗ then
+25
+solve the BD subproblem using y to obtain Benders cut C
+26
+if C ∈ C then
+27
+go to line 31
+28
+else
+29
+update C ← C ∪ C and refine D w.r.t. C
+30
+go to line 23
+31
+forall u in the last exact layer of D do
+32
+update ˆY ← ˆY ∪ {˜y} where ˜y encodes longest r-u path of D
+33
+go to line 2
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+11
+The algorithm starts with constructing a restricted DD D corresponding to MC(ˆy) with empty
+initial values for C and ˆy. We then find a longest r-t path of D encoding solution (y,z). Next,
+using y, we solve the associated subproblem to obtain a feasibility/optimality cut C. We add this
+cut to C, refine D according to it, and find a new longest r-t path. We repeat these steps until no
+new feasibility/optimality cut is generated. At this point, the length of a longest r-t path of D,
+denoted by w, gives a lower bound to the master problem M, which is also a valid lower bound
+to the original problem H. The value of w can be used to update w∗, the optimal value of H
+at termination. Next, we create a relaxed DD D corresponding to MC(ˆy). We find a longest r-t
+path of D that provides an upper bound w to M. If the upper bound w is strictly greater than
+the current value of w∗, we follow steps similarly to the case for D to iteratively refine D w.r.t.
+feasibility/optimality cuts through solving the subproblems, until no new cut is generated. Next,
+we perform a specialized branch-and-bound procedure to improve the bound through expanding
+merged layers of the DD. To this end, we add all the partial assignments associated with nodes in
+the last exact layer of D (the last node layer in which no nodes are merged) to the collection ˆY.
+The nodes corresponding to partial assignments in ˆY are required to be further explored to check
+whether or not the value of w∗ can be improved. That is, the above process is repeated for every
+node v with partial assignment in ˆY as the r-v path is fixed in the new restricted/relaxed DDs.
+The algorithm terminates when ˆY becomes empty, at which point w∗ is the optimal value.
+3.
+DD-BD Formulation for the SGUFP
+In this section, we adapt the DD-BD framework described in Section 2.2 to solve the SGUFP.
+3.1.
+MIP Formulation
+We study the MIP formulation of the SGUFP based on that of its deterministic counterpart given
+in Davarnia et al. (2019). Consider a network G = (V,A) with node set V := V ′ ∪ {s,t} and arc set
+A, where s and t are source and sink nodes, respectively. The source node is connected to all the
+supply nodes in S ⊆ V ′, and the sink node is connected to all the demand nodes in D ⊆ V ′. Figure 4
+illustrates the general structure of this network. For a node q ∈ V , let δ−(q) := {i ∈ V | (i,q) ∈ A}
+and δ+(q) := {j ∈ V | (q,j) ∈ A} show the set of incoming and outgoing neighbors of q, respectively.
+Define ¯V ⊆ V ′ as a subset of vertices that must satisfy the NSNM requirement. For each node
+q ∈ ¯V , let binary variable yq
+ij ∈ {0,1} represent whether or not the flow entering node q ∈ ¯V through
+arc (i,q) leaves node q through arc (q,j). The first stage of SGUFP determines the matching pairs
+between incoming and outgoing arcs of unsplittable nodes as follows:
+max
+z
+(1a)
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+12
+S
+D
+s
+t
+Figure 4
+Illustration of network G = (V ′ ∪ {s,t},A)
+s.t.
+�
+j∈δ+(q)
+yq
+ij ≤ 1
+∀i ∈ δ−(q), ∀q ∈ ¯V
+(1b)
+�
+i∈δ−(q)
+yq
+ij ≤ 1
+∀j ∈ δ+(q), ∀q ∈ ¯V
+(1c)
+yq
+ij ∈ {0,1}
+∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V,
+(1d)
+where constraints (1b) ensure that each incoming arc to a node with NSNM requirement is
+assigned to at most one outgoing arc, and constraints (1c) guarantee that each outgoing arc from
+such a node is matched with at most one incoming arc.
+In (1a)–(1d), variable z represents the objective value of the second stage of SGUFP where
+the demand uncertainty is taken into account. This demand uncertainty is modeled by a set Ξ of
+scenarios for the demand vector dξ with occurrence probability Prξ for each scenario ξ ∈ Ξ. Let
+continuous variable xξ
+ij ∈ R+ denote the flow from node i to node j through arc (i,j) under scenario
+ξ ∈ Ξ. We further assign a reward rij per unit flow to be collected by routing flow through arc (i,j).
+It follows that z = �
+ξ∈Ξ Prξzξ, where zξ is the objective value of the second stage of SGUFP for
+each scenario ξ ∈ Ξ. This subproblem is formulated as follows for a given y vector:
+max
+�
+q∈V
+�
+j∈δ+(q)
+rqjxξ
+qj
+(2a)
+s.t.
+�
+i∈δ−(q)
+xξ
+iq −
+�
+j∈δ+(q)
+xξ
+qj = 0
+∀q ∈ V ′
+(2b)
+ℓξ
+iq ≤ xξ
+iq ≤ uξ
+iq
+∀i ∈ δ−(q), ∀q ∈ V
+(2c)
+xξ
+iq − xξ
+qj ≤ uξ
+iq(1 − yq
+ij)
+∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V
+(2d)
+xξ
+qj − xξ
+iq ≤ uξ
+qj(1 − yq
+ij)
+∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V
+(2e)
+xξ
+iq ≤ uξ
+iq
+�
+j∈δ+(q)
+yq
+ij
+∀i ∈ δ−(q), ∀q ∈ ¯V
+(2f)
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+13
+xξ
+qj ≤ uξ
+qj
+�
+i∈δ−(q)
+yq
+ij
+∀j ∈ δ+(q), ∀q ∈ ¯V
+(2g)
+xξ
+ij ≥ 0
+∀(i,j) ∈ A.
+(2h)
+In the above formulation, the objective function captures the total reward collected by routing
+flows throughout the network (from the source s to the sink t) to satisfy demands. The flow-balance
+requirements are represented by (2b). Constraints (2c) bound the flow on each arc from below
+and above. To impose the demand requirement for each scenario ξ ∈ Ξ, we fix ℓξ
+qt = uξ
+qt = dξ
+q for all
+demand nodes q ∈ D with demand dξ
+q, and leave the lower and upper bound values unchanged for all
+other arcs. Constraints (2d)–(2g) model the NSNM requirement for each node q ∈ ¯V . In particular,
+(2d) and (2e) ensure that matching arcs (i,q) and (q,j) have equal flows. Constraints (2f) and (2g)
+guarantee that an arc without a matching pair does not carry any flow. We note here that the
+Constraint (2b) is implied by other constraints of the above subproblem under the assumption
+that y is feasible to the master problem (1a)–(1d). However, we maintain this constraint in the
+subproblem because the master formulation in our DD-based approach, as will be described in
+Section 3.2, may produce a solution that is not feasible to (1a)–(1d). As a result, the addition of
+the Constraint (2b) will lead to a tighter subproblem formulation.
+As discussed in Section 2.2, the first step to use the DD-BD algorithm is to decompose the
+underlying problem into a master and a subproblem. The above two-stage formulation of the
+SGUFP is readily amenable to BD since the first stage problem (1a)-(1d) can be considered as the
+master problem together with some valid lower and upper bounds −Γ and Γ on z induced from the
+boundedness of the MIP formulation. For a given y value obtained from the master problem and
+a scenario ξ ∈ Ξ, the second stage problem (2a)-(2h) can be viewed as the desired subproblems.
+The optimality/feasibility cuts obtained from each scenario-based subproblem are then added to
+the master problem through aggregation as described in Section 3.3.
+3.2.
+DD-BD: Master Problem Formulation
+While the DD-BD Algorithm 1 provides a general solution framework for any bounded MIP, its DD
+component is problem-specific, i.e., it should be carefully designed based on the specific structure
+of the underlying problem. In this section, we design such an oracle for the SGUFP that represents
+the feasible region {(1b)−(1d),z ∈ [−Γ,Γ]} of the master problem (1a)-(1d). To model this feasible
+region in the original space of (y;z) variables, a DD would require �
+q∈ ¯V |δ−(q)| × |δ+(q)| arc
+layers to represent binary variables y and one arc layer to encode the continuous variable z.
+Constructing such a DD, however, would be computationally cumbersome due to the large number
+of the arc layers. To mitigate this difficulty, we take advantage of the structural flexibility of DDs
+in representing irregular variable types that cannot be used in standard MIP models. One such
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+14
+variable type is the index set, where arc layers represent indices, rather than domain values. We
+next show that we can remarkably reduce the number of DD arc layers by reformulating the master
+problem in a transformed space of variables defined over index sets.
+Consider a node q ∈ ¯V . In the following, we define mappings that assign an index to each incoming
+and outgoing arc of q. These mappings enable us to define new variables to reduce the number
+of DD arc layers. Let ind−(i,q) be a one-to-one mapping from incoming arcs (i,q), for i ∈ δ−(q),
+to the index set {1,2,...,|δ−(q)|}. Similarly, let ind+(q,j) be a one-to-one mapping from outgoing
+arcs (q,j), for j ∈ δ+(q), to the index set {1,2,...,|δ+(q)|}. For each incoming arc (i,q) with index
+h = ind−(i,q), we define an integer variable wq
+h ∈ {0,1,...,|δ+(q)|} such that wq
+h = 0 if this incoming
+arc is not paired with any outgoing arc, and wq
+h = k > 0 if this arc is matched with an outgoing arc
+(q,j) with index k = ind+(q,j).
+Next, we give a formulation in the space of w variables that describes the matching between
+incoming and outgoing arcs of q for all q ∈ ¯V . In the following, sign(.) represents the sign function
+that returns 1 if its argument is strictly positive, 0 if the argument is zero, and −1 otherwise.
+Further, the operator |.|, when applied on a set, represents the set size; and when applied on a real
+number, it represents the absolute value.
+Proposition 2. Formulation
+�
+i∈δ−(q)
+sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≥
+��δ−(q)
+�� − 1
+∀j ∈ δ+(q), ∀q ∈ ¯V
+(3a)
+wq
+ind−(i,q) ∈
+�
+0,1,...,
+��δ+(q)
+���
+∀i ∈ δ−(q), ∀q ∈ ¯V
+(3b)
+models the matching between incoming and outgoing arcs of nodes q ∈ ¯V .
+Proof.
+We show the result for a single node q ∈ ¯V . The extension to the multiple node case
+is straightforward as the matching problem for each node is independent from other nodes. For
+the direct implication, assume that M q is a matching between incoming and outgoing arcs of q,
+with elements of the form (i,j) that represent a matching between the incoming arc (i,q) and
+the outgoing arc (q,j). We show that variables w associated with matching pairs in M q satisfy
+constraints (3a) and (3b). It follows from the definition of w that, for each (i,j) ∈ M q, we have
+wq
+ind−(i,q) = ind+(q,j). Also, for any i ∈ δ−(q) that does not have a matching pair in M q, we have
+wq
+ind−(i,q) = 0. These value assignments show that w satisfies (3b) as the image of ind+ mapping is
+{1,...,|δ+(q)|}. For each i ∈ δ−(q) and j ∈ δ+(q), we have
+���wq
+ind−(i,q) − ind+(q,j)
+��� ≥ 0, with equality
+holding when (i,j) ∈ M q. For each j ∈ δ+(q), there are two cases. For the first case, assume that
+(i,j) /∈ M q for any i ∈ δ−(q). As a result,
+���wq
+ind−(i,q) − ind+(q,j)
+��� > 0 for all i ∈ δ−(q). Applying
+the sign(.) function on these terms yields sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+= 1, which implies that
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+15
+�
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+= |δ−(q)|, satisfying (3a). For the second case, assume that
+(i∗,j) ∈ M q for some i∗ ∈ δ−(q). As a result, we have �
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+=
+|δ−(q)| − 1 since sign
+����wq
+ind−(i∗,q) − ind+(q,j)
+���
+�
+=
+���wq
+ind−(i∗,q) − ind+(q,j)
+��� = 0, satisfying (3a).
+For the reverse implication, assume that w is a feasible solution to (3a)–(3b). We show that the
+pairs of the form (i,j) encoded by these variables constitute a feasible matching between incoming
+and outgoing arcs of q, i.e., (i) each arc (i,q) is matched with at most one arc (q,j), and (ii) each
+arc (q,j) is matched with at most one arc (i,q). It follows from constraint (3b) that, for each i ∈
+δ−(q), variable wq
+ind−(i,q) takes a value between {0,1,...,|δ+(q)|}. If wq
+ind−(i,q) = 0, then (i,q) is not
+matched with any outgoing arc, otherwise it is matched with arc (q,j) with ind+(q,j) = wq
+ind−(i,q).
+This ensures that condition (i) above is satisfied for this matching collection. Further, for each
+j ∈ δ−(q), constraint (3a) implies that sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+can be equal to zero for at
+most one i ∈ δ−(q). In such a case, we would have at most one matching pair of the form (i,j) in
+the collection, showing that condition (ii) above is satisfied.
+□
+It follows from Proposition 2 that constraints (3a)-(3b) can replace (1b)-(1d) in the master
+problem (1a)-(1d) to obtain the following master problem in a transformed space of variables.
+max
+w;z {z | (3a) − (3b),z ∈ [−Γ,Γ]}.
+(4)
+Note that formulation (4) is an integer nonlinear program (INLP) with nonconvex and non-
+continuous constraint functions. Such a formulation is extremely difficult for conventional MINLP
+techniques and solvers to handle. However, due to structural flexibility of DDs in representing inte-
+ger nonlinear programs, this problem can be easily modeled via a DD; see Davarnia and Van Hoeve
+(2020) for a detailed account on using DDs for modeling INLPs. In the following, we present an
+algorithm to construct DDs in the space of (w;z) variables for the master problem (4) with a
+single node q ∈ ¯V . The extension to the case with multiple nodes follows by replicating the DD
+structure. The output of Algorithm 2 is a DD with |δ−(q)| + 1 arc layers where the first |δ−(q)|
+layers represent w variables and the last layer encodes variable z. In this algorithm, su denotes the
+state value of DD node u. The core idea of the algorithm is to use unpaired outgoing arcs of q as
+the state value at each DD layer that represents the matching for an incoming arc of q.
+Next, We show that the solution set of the DD constructed by Algorithm 2 represents the feasible
+region of (4). Note here that DD representation of a MIP set, as described in Section 2.2, does
+not imply the encoding of all of the solutions of the set, but rather the encoding of a subset of all
+solutions that subsumes all the extreme points of the set. Such a representation is sufficient to solve
+an optimization problem over the set with an objective function convex in continuous variables,
+which is the case for (4).
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+16
+Algorithm 2: Construction of DD for the master problem of SGUFP with a node q ∈ ¯V
+Data: node q ∈ ¯V , parameter Γ
+Result: an exact DD D
+1 create the root node r ∈ U1 with state sr = {0,1,...,|δ+(q)|}
+2 forall i ∈ {1,2,...,|δ−(q)|} and u ∈ Ui do
+3
+forall ℓ ∈ su do
+4
+create a node v ∈ Ui+1 with state (su \ {ℓ}) ∪ {0} and an arc a ∈ Ai connecting u to v
+with label l(a) = ℓ
+5 forall u ∈ U1+|δ−(q)| do
+6
+create two arcs a1,a2 ∈ A1+|δ−(q)| connecting u to the terminal node with labels l(a1) = Γ
+and l(a2) = −Γ.
+Theorem 1. Consider a SGUFP with ¯V = {q}. Let D be a DD constructed by Algorithm 2.
+Then, Sol(D) represents the feasible region of (4).
+Proof.
+(⊆) Consider an r-t path of D that encodes solution ( ˜wq,z). According to Algorithm 2,
+the labels of the first |δ−(q)| arcs of this path belong to {0,1,...,|δ+(q)|}, showing that ˜wq
+satisfies constraints (3b). Assume by contradiction that ˜wq does not satisfy constraints (3a),
+i.e., �
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≤ |δ−(q)| − 2 for some j ∈ δ+(q). This implies that
+˜wq
+ind−(i′,q) = ˜wq
+ind−(i′′,q) = ind+(q,j) for two distinct i′,i′′ ∈ δ−(q). In other words, the arcs at lay-
+ers ind−(i′,q) and ind−(i′′,q) of the selected r-t path both share the same label value ind+(q,j).
+According to line 3 of Algorithm 2, we must have that the state value of nodes at layers ind−(i′,q)
+and ind−(i′′,q) of the r-t path both contain ind+(q,j). This is a contradiction to the state update
+policy in line 4 of Algorithm 2, since positive arc labels at each layer of the DD will be excluded
+from the state value of the subsequent nodes.
+(⊇) Consider a feasible solution point ( ˜wq; ˜z) of (4). Suppose ˜wq = (ℓ1,ℓ2,...,ℓ|δ−(q)|). According
+to constraints (3a), no two coordinates of ˜wq have the same positive value. The state value at the
+root node in D contains all index values {0,1,...,|δ+(q)|}. According to Algorithm 2, there exists
+an arc with label ℓ1 at the first layer of D. The state value at the head node of this arc, therefore,
+contains ℓ2 ∈ {0,1,...,|δ+(q)|} \ {ℓ1}, which guarantees an arc with label ℓ2 at the second layer of
+this path. Following a similar approach, we can track a path from the root to layer |δ−(q)| whose
+arcs labels match values of ˜wq. Note for the last layer that ˜z ∈ [−Γ,Γ], which is included in the
+interval between arc labels of the last layer of D. As a result, ( ˜wq; ˜z) is represented by an r-t path
+of D.
+□
+The main purpose of using a DD that models the master problem (4) over one that models (1a)-
+(1d) is the size reduction in arc layers that represent variables w as compared with variables
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+17
+y. It turns out that this space transformation can significantly improve the solution time of the
+DD approach. We refer the interested reader to Appendix A for a detailed discussion on these
+advantages, including preliminary computational results.
+Constructing exact DDs as described in Algorithm 2 can be computationally expensive for large
+size problems. As discussed in Section 2.2, relaxed and restricted DDs are used to circumvent this
+difficulty. Building restricted DDs is straightforward as it involves the selection of a subset of r-t
+paths of the exact DD that satisfy a preset width limit. Constructing relaxed DDs, on the other
+hand, requires careful manipulation of the DD structure to merge nodes in such a way that it
+encodes a superset of all r-t paths of the exact DD. We demonstrate a method to construct such
+relaxed DDs in Algorithm 3. Similarly to Algorithm 2, this algorithm is presented for a single
+NSNM node, but can be extended to multiple nodes by replicating the procedure.
+Algorithm 3: Construction of relaxed DD for the master problem of SGUFP with a node
+q ∈ ¯V
+Data: node q ∈ ¯V , parameter Γ
+Result: a relaxed DD D
+1 create the root node r ∈ U1 with state sr = {0,1,...,|δ+(q)|}
+2 forall i ∈ {1,2,...,|δ−(q)|} and u ∈ Ui do
+3
+forall ℓ ∈ su do
+4
+create a node v ∈ Ui+1 with state (su \ {ℓ}) ∪ {0} and an arc a ∈ Ai connecting u to v
+with label l(a) = ℓ
+5
+select a subset of nodes v1,v2,...,vk ∈ Ui+1 and merge them into node v′ with state
+sv′ = �k
+j=1 svj
+6 forall u ∈ U1+|δ−(q)| do
+7
+create two arcs a1,a2 ∈ A1+|δ−(q)| connecting u to the terminal node with labels l(a1) = Γ
+and l(a2) = −Γ.
+Theorem 2. Consider a SGUFP with ¯V = {q}. Let D be a DD constructed by Algorithm 3.
+Then, D represents a relaxation of the feasible region of (4).
+Proof.
+Let ˙D be the DD constructed by Algorithm 2 for the master problem (4) with a single
+node q ∈ ¯V . It suffices to show that the solution set of D provides a relaxation for that of ˙D. Pick
+a root-terminal path ˙P of ˙D with encoding point ( ˙wq; ˙z). We show that there exist a root-terminal
+path P of D with encoding point (wq;z) such that wq = ˙wq and z = ˙z. Given a DD, define Pk to
+be a sub-path composed of arcs in the first k layers, for 1 ≤ k ≤ |δ−(q)|. We show for any sub-path
+˙Pk of ˙D with encoding point ˙wq
+k = ( ˙wq
+1,..., ˙wq
+k), there exists a sub-path P k of D with encoding
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+18
+point wk = (w1,...,wk) such that wh = ˙wh for h = 1,...,k. Note that we only need to prove the
+matching values for k ≤ |δ−(q)|, because each node at node layer |δ−(q)| + 1 of both ˙D and D
+is connected by two arcs with labels −Γ and Γ to the terminal node, and thus there are always
+matching arcs with the same label for the last layer, i.e., z = ˙z. We prove the result by induction on
+k. The base case for k = 1 is trivial, since D contains arcs with labels {0,1,...,|δ+(q)|} in the first
+layer, which includes the label value of the first arc on ˙P1. For the induction hypothesis, assume
+that the statement is true for k = d, i.e., for the sub-path ˙Pd with label values ˙wq
+d = ( ˙wq
+1,..., ˙wq
+d),
+there is sub-path P d of D with matching arc labels. We show the statement holds for d + 1. Let
+u ∈ ˙Ad+1 and v ∈ Ad+1 be the end nodes of ˙Pd and P d, respectively. It follows from Algorithm 2
+that the index set representing the state value at node u contains ˙wq
+d+1, i.e., ˙wq
+d+1 ∈ ˙su = {0} ∪
+{1,...,|δ+(q)|} \ { ˙w1, ˙w2,..., ˙wd}. The merging step in line 5 of Algorithm 3, on the other hand,
+implies that sv ⊇ {0}∪{1,...,|δ+(q)|}\{w1,w2,...,wd} = {0}∪{1,...,|δ+(q)|}\{ ˙w1, ˙w2,..., ˙wd} =
+˙su, where the inclusion follows from the fact that state values at nodes on path P d contain those of
+each individual path due to merging operation, and the first equality holds because of the induction
+hypothesis. As a result, sv must contain ˙wq
+d+1, which implies that there exists an arc with ˙wq
+d+1
+connected to node v on P d. Attaching this arc to P d, we obtain the desired sub-path P d+1.
+□
+3.3.
+DD-BD: Subproblem Formulation
+At each iteration of the DD-BD algorithm, an optimal solution of the master problem is plugged into
+the subproblems to obtain feasibility/optimality cuts. For the SGUFP formulation, this procedure
+translates to obtaining an optimal solution of (4) in the space of w variables, which is used to
+solve the subproblem (2a)-(2h). The formulation of the subproblem, however, is defined over the
+original binary variables y, and the resulting feasibility/optimality cuts are generated in this space.
+To remedy this discrepancy between the space of variables in the master and subproblems, we need
+to find a one-to-one mapping between variables w and y, as outlined next.
+Proposition 3. Consider a node q ∈ ¯V . Let yq be a feasible solution to (1b)-(1d). Then, wq
+obtained as
+wq
+ind−(i,q) =
+�
+j∈δ+(q)
+ind+(q,j)yq
+ij
+∀i ∈ δ−(q),
+(5)
+is a feasible solution to (3a)-(3b). Conversely, let wq be a feasible solution to (3a)-(3b). Then, yq
+obtained as
+yq
+ij = 1 − sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+∀(i,j) ∈ δ−(q) × δ+(q),
+(6)
+is a feasible solution to (1b)-(1d).
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+19
+Proof.
+For the direct statement, let yq be a feasible solution to (1b)-(1d), and construct a
+vector wq according to (5). We show that wq satisfies all constraints (3a)-(3b). First, we show
+that constraints (3a) are satisfied. Assume by contradiction that there exists j′ ∈ δ+(q) such that
+�
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j′)
+���
+�
+≤ |δ−(q)| − 2. This implies that wq
+ind−(i′,q) = wq
+ind−(i′′,q) =
+ind+(q,j′) for some i′,i′′ ∈ δ−(q). Then, we can write that
+wq
+ind−(i′,q) =
+�
+j∈δ+(q)
+ind+(q,j)yq
+i′j = ind+(q,j′) =
+�
+j∈δ+(q)
+ind+(q,j)yq
+i′′j = wq
+ind−(i′′,q),
+where the first and last equalities hold by (5). The second and third equalities in the above chain
+of relations imply that yq
+i′j′ = yq
+i′′j′ = 1, since ind+(q,j′) > 0. This violates constraints (1c), reaching
+a contradiction. Next, we show that constraints (3b) are satisfied. The proof follows directly from
+construction of wq and constraints (1b).
+For the converse statement, let wq be a feasible solution to (3a)-(3b), and construct a vec-
+tor yq according to (6). We show that yq satisfies all constraints (1b)-(1d). To show that each
+constraint (1b) is satisfied, consider i ∈ δ−(q). We can write that
+�
+j∈δ+(q)
+yq
+ij = |δ+(q)| −
+�
+j∈δ+(q)
+sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≤ |δ+(q)| −
+�
+|δ+(q)| − 1
+�
+= 1,
+where the first equality follows from the construction of yq, and the inequality holds by (3b) as
+���wq
+ind−(i,q) − ind+(q,j)
+��� = 0 for at most one index j ∈ δ+(q). To show that each constraint (1c) is
+satisfied, select j ∈ δ+(q). We have
+�
+i∈δ−(q)
+yq
+ij = |δ−(q)| −
+�
+i∈δ−(q)
+sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≤ 1,
+where the equality follows from the construction of yq, and the inequality holds because of con-
+straint (3a). Finally, each constraint (1d) is satisfied due to the fact that 1 − sign(|.|) ∈ {0,1}.
+□
+Proposition 4. Mappings described by (5) and (6) are one-to-one over their respective
+domains.
+Proof.
+Note that the mapping described by (5) is a linear transformation of the form wq = Byq
+with coefficient matrix B ∈ Z|δ−(q)|×(|δ−(q)||δ+(q)|). It is clear from the identity block structure of B,
+that it is full row-rank, since each column contains a single non-zero element while each row has
+at least one non-zero element. As a result, the null space of B is the origin, which implies that
+ˆwq = ˜wq only if ˆyq = ˜yq.
+For the mapping described by (6), let distinct points ˆwq and ˜wq satisfy (3b). Construct vectors
+ˆyq and ˜yq by (6) using ˆwq and ˜wq, respectively. Because ˆwq and ˜wq are distinct, there must
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+20
+exist i ∈ δ−(q) such that ˆwq
+ind−(i,q) ̸= ˜wq
+ind−(i,q). This implies that at least one of these variables, say
+ˆwq
+ind−(i,q), is non-zero. It follows from (3b) that ˆwq
+ind−(i,q) = ind+(q,j′) for some j′ ∈ δ+(q), and that
+ˆwq
+ind−(i,q) ̸= ind+(q,j′). According to (6), we write that ˆyij′ = 1−sign
+���� ˆwq
+ind−(i,q) − ind+(q,j′)
+���
+�
+= 1,
+and that ˜yij′ = 1 − sign
+���� ˜wq
+ind−(i,q) − ind+(q,j′)
+���
+�
+= 0, showing that ˆyq ̸= ˜yq.
+□
+Using the results of Propositions 3 and 4, we can apply the DD-BD Algorithm 1 in its entirety for
+the SGUFP. In particular, at each iteration of the algorithm, we can transform the optimal solution
+( ¯w, ¯z) obtained from the DD representing the master problem (4) into a solution (¯y, ¯z) through the
+mapping (6). Given an optimal first-stage solution ¯y, we can solve |Ξ| separate subproblems; one
+for each demand realization in the second-stage. The feasibility cuts obtained from subproblems,
+which are in the space of y variables, are translated back into the space of w variables through the
+mapping (5) and added to the master problem. Further, in a case where all subproblems produce
+an optimality cut, they can be aggregated to generate an optimality cut in the space of (y,z),
+which is added to the master problem after being translated into the space of (w,z) variables. The
+master DD will be refined with respect to the resulting inequalities, and an optimal solution is
+returned to be used for the next iteration.
+In the remainder of this section, we present details on the derivation of optimality/feasibility cuts
+from subproblem (2a)-(2h). Consider the following partitioning of the set of arcs A into subsets
+A1 :=
+�
+(i,j) ∈ A
+�� δ−(i) = ∅, δ+(j) ̸= ∅
+�
+, A2 :=
+�
+(i,j) ∈ A
+�� δ−(i) ̸= ∅, δ+(j) = ∅
+�
+,
+A3 :=
+�
+(i,j) ∈ A
+�� δ−(i) ̸= ∅, δ+(j) ̸= ∅
+�
+, A4 :=
+�
+(i,j) ∈ A
+�� δ−(i) = ∅, δ+(j) = ∅
+�
+,
+and let θξ = (βξ,γξ,δξ,φξ,λξ,µξ) be the vector of dual variables associated with constraints of
+(2a)-(2h) for a scenario ξ ∈ Ξ. Further, define the bi-function
+f(y;θξ) =
+�
+q∈V
+�
+j∈δ+(q)
+�
+−ℓqjβξ
+qj + uqjγξ
+qj
+�
++
+�
+q∈ ¯V
+�
+(i,j)∈δ−(q)×δ+(q)
+�
+uiq(1 − yq
+ij)λξ
+iqj + uqj(1 − yq
+ij)µξ
+iqj
+�
++
+�
+q∈ ¯V
+�
+i∈δ−(q)
+�
+�uiq
+�
+j∈δ+(q)
+yq
+ijσξ
+iq
+�
+� +
+�
+q∈ ¯V
+�
+j∈δ+(q)
+�
+�uqj
+�
+i∈δ−(q)
+yq
+ijφξ
+qj
+�
+�.
+For a given ¯y and each scenario ξ ∈ Ξ, the dual of the subproblem (2a)-(2h) can be written as
+follows where the symbol ⋆ on a node means that it belongs to ¯V .
+min
+f(¯y;θξ)
+(7a)
+s.t.
+αξ
+⋆q − βξ
+i⋆q + γξ
+i⋆q +
+�
+j:j∈δ+(⋆q)
+λξ
+i⋆qj −
+�
+j:j∈δ+(⋆q)
+µξ
+i⋆qj + σξ
+i⋆q ≥ ri⋆q
+∀(i,
+⋆q) ∈ A1 (7b)
+αξ
+q − βξ
+iq + γξ
+iq ≥ riq
+∀(i,q) ∈ A1 (7c)
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+21
+− αξ
+⋆q − βξ
+⋆qj + γξ
+⋆qj −
+�
+i:i∈δ−(⋆q)
+λξ
+i⋆qj +
+�
+i:i∈δ−(⋆q)
+µξ
+i⋆qj + φξ
+⋆qj ≥ r⋆qj
+∀(
+⋆q,j) ∈ A2 (7d)
+− αξ
+q − βξ
+qj + γξ
+qj ≥ rqj
+∀(q,j) ∈ A2 (7e)
+− αξ
+⋆q + αξ
+⋆
+j − βξ
+⋆q
+⋆
+j + γξ
+⋆q
+⋆
+j +
+�
+i∈δ−(⋆q)
+�
+µξ
+i⋆q
+⋆
+j − λξ
+i⋆q
+⋆
+j
+�
++
+�
+i∈δ+(
+⋆
+j)
+�
+λξ
+⋆q
+⋆
+ji − µξ
+⋆q
+⋆
+ji
+�
++ σξ
+⋆q
+⋆
+j + φξ
+⋆q
+⋆
+j ≥ r⋆q
+⋆
+j
+∀(
+⋆q,
+⋆j) ∈ A3
+(7f)
+− αξ
+⋆q + αξ
+j − βξ
+⋆qj + γξ
+⋆qj +
+�
+i∈δ−(⋆q)
+�
+µξ
+i⋆qj − λξ
+i⋆qj
+�
++ φξ
+⋆qj ≥ r⋆qj
+∀(
+⋆q,j) ∈ A3 (7g)
+− αξ
+q + αξ
+⋆
+j − βξ
+q
+⋆
+j + γξ
+q
+⋆
+j +
+�
+i∈δ+(
+⋆
+j)
+�
+λξ
+q
+⋆
+ji − µξ
+q
+⋆
+ji
+�
++ σξ
+q
+⋆
+j ≥ rq
+⋆
+j
+∀(q,
+⋆j) ∈ A3 (7h)
+− αξ
+q + αξ
+j − βξ
+qj + γξ
+qj ≥ rqj
+∀(q,j) ∈ A3
+(7i)
+− βξ
+iq + γξ
+iq ≥ riq
+∀(i,q) ∈ A4
+(7j)
+αξ
+q ∈ R
+∀q ∈ V ′ (7k)
+βξ
+ij, γξ
+ij, σξ
+ij, φξ
+ij, λξ
+iqj, µξ
+iqj ≥ 0
+∀i,q,j ∈ V.
+(7l)
+If the above problem has an optimal solution ˆθξ for all ξ ∈ Ξ, the output of the subproblems will
+be an optimality cut of the form �
+ξ∈Ξ Prξf(y; ˆθξ) ≥ z. If the above problem is unbounded along a
+ray ˆθξ for a ξ ∈ Ξ, the output of the subproblem will be a feasibility cut of the form f(y; ˆθξ) ≥ 0.
+Note that replacing variables y in the above constraints with w through the mapping (5) results
+in separable nonlinear constraints. Nevertheless, since these constraints will be used to refine the
+master DD, their incorporation is simple due to structural flexibility of DDs in modeling such
+constraints; we refer the reader to Davarnia and Van Hoeve (2020) for a detailed account for
+modeling INLPs with DDs.
+4.
+Computational Experiments
+In this section, we solve SGUFP as a core model for the unit train scheduling problem with demand
+stochasticity using three different approaches: (i) the standard MIP formulation that is a deter-
+ministic equivalent of the two-stage model and contains all variables and constraints of the master
+problem and |Ξ| subproblems; (ii) the Benders reformulation presented in Section 3.1 composed
+of the master problem (1a)-(1d) and |Ξ| subproblems (2a)-(2h); and (iii) the DD-BD algorithm
+proposed in the present paper. In the Benders approach, we solve separate subproblems using a
+fixed vector ¯y obtained from the master problem. The feasibility cuts generated by subproblems
+are added directly to the constraint set of the master problem, and the optimality cuts are added
+as an aggregated cut over all scenarios. We note here that when there is a feasibility cut for any
+scenario, we add it directly to separate the solution of the current iteration and move on to the
+next iteration. To obtain a valid inequality that provides a bound for the single z variable, we need
+to aggregate valid inequalities over all scenario subproblems as z is composed of the objective value
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+22
+of all these subproblems. Therefore, we can only produce an optimality cut for the z variable when
+we have optimality cuts for all of the subproblems. For the DD-BD approach, we use the following
+algorithmic choices to build restricted and relaxed DDs. For the restricted DDs, we choose a subset
+of the r-t paths with largest lengths, which are more likely to contain an optimal solution. For
+the relaxed DDs, we merge nodes that have the largest number of common members in their state
+values. We refer the reader to Bergman et al. (2016a) for other heuristic approaches that could be
+used for this purpose.
+4.1.
+Test Instances
+In our experiments, we consider the structure of the SGUFP network given in Section 3.1. To
+ensure that the problem is always feasible, we create an artificial node s0 to compensate for any
+shortage of the supply, and add an arc from the artificial supply s0 to each demand node.
+We create test instances based on the specification given in Davarnia et al. (2019), which is
+inspired by realistic models. In particular, we consider a base rail network G′ = (V ′,A′) where 10%
+and 30% of the nodes are supply and demand nodes, respectively. We assume that 50% of the
+nodes must satisfy the NSNM requirement. We then create a network G = (V,A) by augmenting
+supply/demand and artificial nodes as described above with the following settings. The integer
+supply value at supply nodes is randomly selected from the interval [100,600]. The capacity of arcs
+connecting s0 to demand nodes are considered to be unbounded, and the integer capacity value
+of other arcs is randomly selected from the interval [100,300]. For each demand scenario ξ ∈ Ξ,
+the integer demand value at demand nodes is randomly chosen from the interval [100,200]. The
+reward of the arcs connecting s0 to the demand nodes are generated from the interval [−10,−5]
+to represent the cost of lost demands. The reward of the arcs connecting the source to the supply
+nodes is randomly selected from the interval [5,10], and the reward of the arcs connecting the
+demand nodes to the sink is fixed to zero since the flow of these arcs is also fixed. The reward
+of all other arcs is created randomly from the interval [−2,2] where the negative values indicate
+the cost of sending flows through congested arcs. We consider four categories of rail networks with
+|V ′| ∈ {40,60,80,100}. For each category, we create five scenario classes for the number of demand
+scenarios |Ξ| ∈ {50,100,150,200,250}. For each network category and scenario class, we create five
+random instances based on the above settings. Test instances are publicly available (Salemi and
+Davarnia 2022b).
+4.2.
+Numerical Results
+In this section, we present the numerical results that compare the performance of the DD-BD
+formulation for the SGUFP instances with that of the MIP formulation, denoted by “MIP”, and the
+standard Benders reformulation, denoted by “BD”. All experiments are conducted on a machine
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+23
+running Windows 10, x64 operating system with Intel® Core i7 processor (2.60 GHz) and 32 GB
+RAM. The Gurobi optimization solver (version 9.1.1) is used to solve instances for the MIP and
+BD models. When solving problems with Gurobi, we turn off presolve and cuts for all methods
+to have a fair comparison. Tables 1-4 report the running times of each of these formulations for
+|V ′| ∈ {40,60,80,100} and |Ξ| ∈ {50,100,150,200,250} where the time limit is set to 3600 seconds.
+The symbol “ > 3600” indicates that the problem was not solved within the time limit. As evident
+in these tables, the DD-BD formulation outperforms the other alternatives. In particular, the
+gap between the solution time of the DD-BD and the MIP and BD approaches widens as the
+problem size increases. For example, as reported in Table 1, while the DD-BD approach solves all
+25 instances in under 275 seconds, the MIP approach fails to solve 10 of them within 3600 seconds,
+80% of which involve 200 or 250 scenarios. This shows a clear superiority of the DD-BD over the
+MIP method. Further, for most of the instances, the DD-BD approach outperforms the standard
+BD approach, rendering it as the superior solution method among all three. Figures 5-8 compare
+the performance of DD-BD, BD, and MIP formulations through box and whisker plots for each
+network size and under each scenario class. In these figures, for uniformity of illustration, we used
+3600 seconds for the running time of instances that fail to solve the problem within that time
+limit. As the figures show, the minimum, median, and maximum of running times of the DD-BD
+method are remarkably smaller than those of the both BD and MIP methods in all cases. These
+results show the potential of the DD-BD framework in solving network problems with challenging
+combinatorial structures. In Appendix B, we present additional numerical results for the DD-BD
+approach to assess its ability to solve larger problem sizes.
+Table 1
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 40.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+75.74 512.62 2877.19
+> 3600
+> 3600
+BD
+141.83 313.84
+339.81
+451.93
+565.82
+DD-BD
+56.94 129.87
+163.43
+219.02
+274.36
+2
+MIP
+67.59 275.07
+906.10 1892.21 2235.53
+BD
+63.44 121.25
+141.04
+230.81
+235.87
+DD-BD
+42.60
+82.65
+128.16
+164.52
+208.94
+3
+MIP
+94.86 753.23 2453.05
+> 3600
+> 3600
+BD
+71.14 139.20
+172.86
+224.33
+244.91
+DD-BD
+53.32
+93.58
+113.93
+178.65
+217.33
+4
+MIP
+71.46 309.62
+> 3600
+> 3600
+> 3600
+BD
+63.55 182.01
+267.94
+334.74
+380.22
+DD-BD
+46.61
+87.81
+130.19
+183.23
+253.72
+5
+MIP
+380.33 406.73
+> 3600
+> 3600
+> 3600
+BD
+123.69 198.73
+205.16
+231.56
+287.24
+DD-BD
+67.04 104.78
+138.46
+195.69
+231.74
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+24
+Table 2
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 60.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+893.73 > 3600
+> 3600
+> 3600
+> 3600
+BD
+241.85
+556.18
+582.80
+758.54
+933.05
+DD-BD 176.16
+357.06
+603.81
+719.27
+901.02
+2
+MIP
+206.87
+811.64 1554.10
+> 3600
+> 3600
+BD
+259.63
+351.39
+624.08
+816.44 1017.95
+DD-BD 189.07
+388.85
+572.52
+764.76
+961.35
+3
+MIP
+139.70
+702.96 1035.79
+> 3600
+> 3600
+BD
+246.48
+569.37
+628.84
+795.56
+978.15
+DD-BD 142.81
+284.65
+422.52
+565.23
+725.86
+4
+MIP
+153.16
+415.46
+938.03 1681.21 2604.25
+BD
+238.33
+388.19
+563.15
+732.59
+919.08
+DD-BD 131.29
+262.36
+393.18
+521.12
+654.71
+5
+MIP
+165.57
+706.16 2447.15
+> 3600
+> 3600
+BD
+194.12
+244.61
+479.32
+463.63
+617.09
+DD-BD 112.09
+221.30
+332.25
+443.96
+556.33
+Table 3
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 80.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+215.82
+860.21
+> 3600
+> 3600
+> 3600
+BD
+588.51
+806.61 1731.50 1860.12 2051.52
+DD-BD 256.12
+500.52
+757.68 1025.88 1278.13
+2
+MIP
+479.76
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+398.29
+713.01
+861.65 1080.79 1709.04
+DD-BD 184.34
+379.04
+724.66 1088.21 1587.90
+3
+MIP
+238.79
+996.22
+> 3600
+> 3600
+> 3600
+BD
+702.18 1236.58 1650.42 1773.63 2227.89
+DD-BD 285.13
+518.46
+778.97 1046.39 1326.22
+4
+MIP
+404.26 2441.64 2855.29
+> 3600
+> 3600
+BD
+572.83 1219.37 1334.21 1745.91 2089.80
+DD-BD 263.78
+665.30 1230.81 1277.93 1444.02
+5
+MIP
+778.50
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+231.11
+481.31
+625.91 1310.24 1452.27
+DD-BD 187.34
+376.96
+564.34 1205.54 1412.94
+Table 4
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 100.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+774.18
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+1282.59 1728.71 1848.49 2307.74 3309.93
+DD-BD
+698.36 1427.38 1731.95 2014.96 3323.54
+2
+MIP
+480.97
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+781.47 1573.23 1820.79 2672.18 2819.61
+DD-BD
+586.89 1171.96 1848.49 2471.49 2635.22
+3
+MIP
+3071.37
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+1072.14 1322.96 2112.50 2951.55 3412.99
+DD-BD
+485.31
+703.70 1055.36 1803.66 2269.97
+4
+MIP
+838.79 2585.38
+> 3600
+> 3600
+> 3600
+BD
+1548.93 1738.92 2580.53 2616.19 3169.28
+DD-BD
+554.89
+743.64 1098.82 2052.73 3094.23
+5
+MIP
+714.39
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+808.48 1013.68 1722.01 2824.14 3282.10
+DD-BD
+353.48
+700.57 1680.60 2213.81 2907.78
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+25
+Figure 5
+Comparison of DD-BD, BD, and MIP models when |V ′| = 40 under five scenarios
+Figure 6
+Comparison of DD-BD, BD, and MIP models when |V ′| = 60 under five scenarios
+We conclude this section by noting that, while the focus of this paper has been on the unit train
+problem with the no-split no-merge requirements, the proposed DD-BD framework can be applied
+to model network problems that contain additional side constraints on the flow variables, as those
+constraints can be handled in the subproblems while the DD structure in the master problem
+
+4000.00
+3500.00
+3000.00
+Running time (sec)
+2500.00
+2000.00
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+DD-BDBDMIP4000.00
+3500.00
+3000.00
+Running time (sec)
+2500.00
+2000.00
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+IDD-BD
+■BDMIPSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+26
+Figure 7
+Comparison of DD-BD, BD, and MIP models when |V ′| = 80 under five scenarios
+Figure 8
+Comparison of DD-BD, BD, and MIP models when |V ′| = 100 under five scenarios
+remains intact. Examples of such side constraints include the usage-fee limitation (Holzhauser,
+Krumke, and Thielen 2017b) and the flow ratio requirement (Holzhauser, Krumke, and Thielen
+2017a). Applying the DD-BD method to such network models and assessing its effectiveness com-
+pared to alternative approaches could be an interesting direction for future research.
+
+4000.00
+3500.00
+3000.00
+Running time (sec)
+2500.00
+2000.00
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+DD-BDBDMIP4000.00
+3500.00
+3000.00
+time (sec)
+2500.00
+2000.00
+Running 
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+DD-BDBDMIPSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+27
+5.
+Conclusion
+In this paper, we introduce a DD-based framework to solve the SGUFP. This framework uses
+Benders decomposition to decompose the SGUFP into a master problem composed of the combi-
+natorial NSNM constraints, and a subproblem that solves a continuous network flow model. The
+master problem is modeled by a DD, which is successively refined with respect to the cuts generated
+through subproblems. To assess the performance of the proposed method, we apply it to a variant
+of unit train scheduling problem formulated as a SGUFP, and compare it with the standard MIP
+and Benders reformulation of the problem.
+Acknowledgments
+This project is sponsored in part by the Iowa Energy Center, Iowa Economic Development Authority and
+its utility partners. We thank the anonymous referees and the Associate Editor for their helpful comments
+that contributed to improving the paper.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+28
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+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+32
+Appendix A:
+Comparison of Master Problem Formulations
+In this section, we describe the differences between DDs in the space of w variables and those in the space
+of original y in the master problem formulation (4) in Section 3.2. First, we illustrate the size difference
+between these DDs in Example 3.
+Example 3. Consider a directed graph G = (V,A) with node set V = {1,2,q,3,4} and arc set A =
+{(1,q),(2,q),(q,3),(q,4)} where the central node q is subject to NSNM constraints. Let ind−(1,q) =
+ind+(q,3) = 1 and ind−(2,q) = ind+(q,4) = 2. Then, the exact DDs showed in Figures 9(a) and 9(b) with
+three and five arc layers represent the feasible region of master problem (4) and (1a)-(1d), respectively, where
+−M and M are valid bounds for variable z.
+(a) A DD in the space of w variables.
+Numbers next to arcs represent labels.
+(b) A DD in the space of y variables.
+Numbers next to arcs represent labels.
+Figure 9
+Comparison of the number of arc layers for DDs in the space of w and y variables
+As evident from the above example, the main advantage of using a DD in the space of w is the reduction in
+the number of arc layers, which is the main determinant of the DDs computational efficiency. In particular,
+even though such a DD has a larger number of nodes at the layers, a relaxed DD can be constructed to limit
+the width, and hence provide an efficient relaxed DD in a smaller dimension, whereas the relaxations of the
+DD constructed in the space of y variables would still be higher-dimensional.
+To assess the computational efficiency of the solution approach in relation to the DD space, we compare
+the performance of the DD-BD method under two different settings: (i) where DDs are built in the space
+of w variables, denoted by DD-BD-w, and (ii) where DDs are built in the space of y variables, denoted by
+DD-BD-y. We report the results of these two implementations for |V ′| ∈ {40,80} and under five different
+scenarios in Table 5 and Table 6.
+As observed in these tables, the DD-BD-w solves all instances faster than DD-BD-y, with orders of
+magnitude time improvement as the problem size (number of scenarios) increases. These preliminary com-
+putational results show the advantage of designing the DD-BD method for the SGUFP in a transformed
+space of variables.
+
+2
+0
+1
+0
+2
+0
+2
+1
+0
+W
+M
+M
+M
+-M
+M
+M
+I七0
+91,3
+1
+y2,3
+0
+0
+1
+0
+0
+0
+b
+y2,4
+0
+0
+0
+M
+M
+M
+-M
+2
+M
+MSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+33
+Table 5
+Running times (in seconds) of DD-BD-w and DD-BD-y for |V ′| = 40.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD-w
+56.94 129.87 163.43 219.02 274.36
+DD-BD-y
+89.68 304.08 432.34 642.70 839.57
+2
+DD-BD-w
+42.60
+82.65 128.16 164.52 208.94
+DD-BD-y
+68.23 148.76 244.53 344.86 605.04
+3
+DD-BD-w
+53.32
+93.58 113.93 178.65 217.33
+DD-BD-y
+83.05 157.67 310.07 541.33 658.98
+4
+DD-BD-w
+46.61
+87.81 130.19 183.23 253.72
+DD-BD-y
+78.11 149.26 325.31 460.73 694.57
+5
+DD-BD-w
+67.04 104.78 138.46 195.69 231.74
+DD-BD-y
+109.61 223.78 351.80 532.12 669.78
+Table 6
+Running times (in seconds) of DD-BD-w and DD-BD-y for |V ′| = 80.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD-w 256.12
+500.52
+757.68 1025.88 1278.13
+DD-BD-y
+483.42
+977.03 1642.27 3175.72 4230.29
+2
+DD-BD-w 184.34
+379.04
+724.66 1088.21 1587.90
+DD-BD-y
+340.13
+864.21 1856.96 3010.55 4843.67
+3
+DD-BD-w 285.13
+518.46
+778.97 1046.39 1326.22
+DD-BD-y
+568.32 1176.44 2401.98 3326.76 4283.58
+4
+DD-BD-w 263.78
+665.30 1230.81 1277.93 1444.02
+DD-BD-y
+501.04 1430.77 2868.92 3356.39 4356.48
+5
+DD-BD-w 187.34
+376.96
+564.34 1205.54 1412.94
+DD-BD-y
+354.37
+781.18 1279.73 3001.72 3834.08
+Appendix B:
+Additional Computational Experiments
+In this section, we present additional numerical results to assess the limits of the DD-BD method for larger
+problem instances. These results are given in Tables 7 and 8, where the columns are defined similarly to
+those of Tables 1-4. For these instances, the time limit is set to 3600 seconds, and the symbol “> 3600”
+indicates that the problem is not solved within this time limit.
+Table 7
+Running times (in seconds) of DD-BD for |V ′| = 120.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD 1494.49 2824.58
+> 3600 > 3600 > 3600
+2
+DD-BD
+975.47 1892.41 3198.18 > 3600 > 3600
+3
+DD-BD 1150.30 2263.09 3454.47 > 3600 > 3600
+4
+DD-BD 1261.59 2403.79
+> 3600 > 3600 > 3600
+5
+DD-BD
+906.34 1863.15 3050.68 > 3600 > 3600
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+34
+Table 8
+Running times (in seconds) of DD-BD for |V ′| = 150.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD 2496.16 > 3600 > 3600 > 3600 > 3600
+2
+DD-BD 2944.20 > 3600 > 3600 > 3600 > 3600
+3
+DD-BD 2321.62 > 3600 > 3600 > 3600 > 3600
+4
+DD-BD 2590.34 > 3600 > 3600 > 3600 > 3600
+5
+DD-BD 2298.36 > 3600 > 3600 > 3600 > 3600
+

ktE5T4oBgHgl3EQfGw6j/content/tmp_files/2301.05434v1.pdf.txt

@@ -0,0 +1,1103 @@
+LVRNet: Lightweight Image Restoration for
+Aerial Images under Low Visibility
+Esha Pahwa*
+BITS Pilani
+[email protected]
+Achleshwar Luthra*
+Carnegie Mellon University
+[email protected]
+Pratik Narang
+BITS Pilani
+[email protected]
+Abstract
+Learning to recover clear images from images having a
+combination of degrading factors is a challenging task.
+That being said, autonomous surveillance in low visibility
+conditions caused by high pollution/smoke, poor air qual-
+ity index, low light, atmospheric scattering, and haze dur-
+ing a blizzard becomes even more important to prevent ac-
+cidents. It is thus crucial to form a solution that can re-
+sult in a high-quality image and is efficient enough to be
+deployed for everyday use. However, the lack of proper
+datasets available to tackle this task limits the performance
+of the previous methods proposed. To this end, we generate
+the LowVis-AFO dataset, containing 3647 paired dark-hazy
+and clear images. We also introduce a lightweight deep
+learning model called Low-Visibility Restoration Network
+(LVRNet). It outperforms previous image restoration meth-
+ods with low latency, achieving a PSNR value of 25.744 and
+an SSIM of 0.905, making our approach scalable and ready
+for practical use. The code and data can be found here.
+1. Introduction
+Image enhancement and restoration have been a critical
+area of research using both traditional digital image pro-
+cessing techniques[12] [2], and the recent deep learning
+frameworks[32][33][44]. The goal of image restoration is
+to recover a clear image, whereas image enhancement is to
+improve the quality of the degraded image. In this study, we
+perform recovery of the clear image from the hazy version
+while performing low-light image enhancement using a sin-
+gle convolutional network, which could further be applied
+to tasks such as search and rescue operations using object
+detection.
+*equal contribution
+Using deep learning algorithms for image recovery has
+many benefits, the most important being that it can general-
+ize to different variations in the images captured. Hence, we
+observe that deep learning-based methods on most bench-
+mark datasets often outperform traditional methods signif-
+icantly.
+However, there are still challenges that the re-
+searchers have to tackle for image restoration.
+Publicly
+available datasets containing a variety of degrading factors
+that model real-world scenarios are few. Hence, most pre-
+vious works have focused on removing one type of degra-
+dation with a specific intensity level. From the perspective
+of computational complexity, recent deep learning methods
+are computationally expensive, and thus they can’t be de-
+ployed on edge devices. Moreover, image restoration has
+been a long-standing ill-posed research problem, as there
+are infinite mappings between the degraded and the clear
+image. Thus, the existing methods still have room for im-
+provement in finding the correct mapping.
+In this work, we focus on developing an end-to-end
+lightweight deep-learning solution for the image restoration
+task. Our major contributions are listed below:
+• Taking inspiration from Non-linear Activation Free
+Network (NAFNet) [5] and Level Attention Module
+[45], we propose a novel algorithm - Low-Visibility
+Restoration Network (LVRNet), that can effectively re-
+cover high-quality images from degraded images taken
+in poor visual conditions (Figure 1).
+• Due to the lack of available datasets that exhibit a com-
+bination of adverse effects, we generate a new dataset,
+namely LowVis-AFO (abbreviation for Low-Visibility
+Aerial Floating Objects dataset). We use AFO [15] as
+our ground truth dataset and synthesize dark hazy im-
+ages. The data generation process has been elaborated
+in Section 4.1.
+1
+arXiv:2301.05434v1  [cs.CV]  13 Jan 2023
+
+Input 
+Zero-DCE
+SGZNet
+DehazeNet
+StarDCE
+BPPNet
+FFANet
+MSBDN-DFF
+Ground Truth / Reference Image
+Our Result
+Input 
+Zero-DCE
+SGZNet
+DehazeNet
+StarDCE
+BPPNet
+FFANet
+MSBDN-DFF
+Ground Truth / Reference Image
+Our Result
+Figure 1. Visual results on the proposed LowVis-AFO dataset. The method used to obtain each result has been mentioned under the
+image.
+• Benchmarking experiments have been provided on the
+LowVis-AFO dataset to help future researchers for
+quantitative comparison.
+Along with that, LVRNet
+surpasses the results obtained using previous image
+restoration techniques by a significant margin.
+• We perform extensive ablation studies to analyze the
+importance of various loss functions existing in cur-
+rent image restoration research. These experiments are
+discussed in detail in Section 5.
+2. Related Works
+This section highlights the previous work done in the fields
+of image dehazing and low-light image enhancement and
+their limitations.
+2.1. Image Dehazing
+Hazy weather is often seen due to floating particles in
+the environment which degrade the quality of the image
+captured. Therefore, many previous works have tried to
+recover a clear image from the hazy one.
+These works
+can be divided into two methods, ones that rely on prior
+assumptions [17] and the atmospheric scattering model
+(ASM) [31] and the others which use deep learning to solve
+the problem, either by combination with ASM [3][34][36]
+or independently [25][26][33][46][50].
+Conventional
+approaches are physically inspired and apply various types
+of sharp image priors to regularize the solution space.
+However, they exhibit shortcomings when implemented
+with real-world images and videos. For example, the dark
+channel prior method (DCP) [31] does not perform well
+in regions containing the sky. These methods [1][11][24]
+are known to be computationally expensive and require
+2
+
+Pre-processing
+Conv
+Post-processing
+Conv.
+NAF-G1
+NAF-G2
++
+LAN
+NAF-G3
+Stacked Feature
+Maps
+Input Image
+Output Image
+Figure 2. Model architecture of the proposed LVRNet. Starting from the top-left: The input image is passed to the pre-processing
+convolution layers where feature maps are learned and passed to NAF Groups (here we have used 3 groups). The features extracted from
+each group are concatenated (or stacked) along the channel dimension and sent as input to the Level Attention Module (LAM). Finally, we
+pass LAM’s output to CNN layers for post-processing, adding the original image through residual connection and extracting the restored
+image at the bottom-left.
+heuristic parameter-tuning. Supervised dehazing methods
+can be divided into two subparts, one is ASM based, and
+the other is non-ASM based.
+ASM-based Learning: MSCNN[34] solves the task of
+image dehazing by dividing the problem into three steps:
+using CNN to estimate the transmission map t(x), using
+statistical methods to find atmospheric light A and then
+recover the clear image J(x) using t(x) and A jointly. Meth-
+ods like LAP-Net [23] adopt the relation of depth with the
+amount of haze in the image. The farther the scene from the
+camera, the denser the haze would be. Hence it considers
+the difference in the haze density in the input image using a
+stage-wise loss, where each stage predicts the transmission
+map from mild to severe haze scenes.
+DehazeNet [3]
+consists of four sequential operations: feature extraction,
+multi-scale mapping, calculating local extremum, and
+non-linear regression. MSRL-DehazeNet [43] decomposes
+the problem into recovering high-frequency and basic com-
+ponents. GCANet [4] employs residual learning between
+haze-free and hazy images as an optimization objective.
+End-to-end Learning: This subpart of previous work cor-
+responds to non-ASM-based deep learning methods for re-
+covering the clear image. Back-Projected Pyramid Network
+(BPPNet) [39] is a generative adversarial network that in-
+cludes iterative blocks of UNets [37] to learn haze features
+and pyramid convolution to preserve spatial features of dif-
+ferent scales. The reason behind using iterative blocks of
+UNets[37] is to avoid increasing the number of encoder lay-
+ers in a single UNet[37] as it leads to a decrease in height
+and width of latent feature representation hence resulting in
+loss of spatial information. Moreover, different blocks of
+UNet learn different complexities of haze features, and the
+final concatenation step ensures that all of them are taken
+into account during image reconstruction. The final recon-
+struction is done using the pyramid convolution block. The
+output feature is post-processed to get a haze-free image.
+Feature-Fusion Attention Network (FFANet) [33] adopts
+the idea of an attention mechanism and skip connections
+to restore haze-free images. A combination of channel at-
+tention and pixel attention is introduced, which helps the
+network, deal with the uneven spatial distribution of haze
+and different weighted information across channels. Au-
+toencoders [6], hierarchical networks [9], and dense block
+networks [14] has also been proposed for the task of image
+dehazing. However, our main comparison lies with FFANet
+[33], wherein we show a huge improvement compared to
+the former method with a model containing a lesser number
+of parameters and which can generalize to different levels
+of haze.
+2.2. Low-light Enhancement
+Traditional methods for low-light image enhancement
+(LLIE) include Histogram Equalization-based methods and
+Retinex model-based methods. Recent research has been
+focused on developing deep learning-based methods fol-
+lowing the success of the first seminal work. Deep learning-
+based solutions are more accurate, robust, and have a
+shorter inference time thus attracting more researchers.
+3
+
+Layer Norm
+1x1, conv
+3x3, dconv
+Simple Gate
+SCA
+1x1, conv
++
+Layer Norm
+1x1, conv
+Simple Gate
+1x1, conv
++
+I/P
+O/P
+NAF BLOCK
+NAF BLOCK
+NAF BLOCK
+CONV
++
+NAF-Group
+NAF Block
+Figure 3. Architecture of NAF Block and NAF Group. NAF Blocks are the building blocks of NAF Groups. A detailed description has
+been provided in Section 3.1 and Section 3.1.1
+Learning strategies used in these methods are mainly su-
+pervised learning [27, 29, 30, 35, 51, 28, 41], unsupervised
+learning [20], and zero-shot learning [49, 13].
+Supervised Learning: The first deep learning-based LLIE
+method LLNet [27] is an end-to-end network that employs a
+variant of stacked-sparse denoising autoencoder to brighten
+and denoise low-light images simultaneously. LLNet in-
+spired many other works [29, 30, 35, 51], but they do not
+consider the observation that noise exhibits different lev-
+els of contrast in different frequency layers. Later, Xu et
+al. [41] proposed a network that suppresses noise in the
+low-frequency layers and recovers the image contents by
+inferring the details in high-frequency layers. There is an-
+other division of methods that is based on the Retinex the-
+ory. Deep Retinex-based models [40, 42] decomposes the
+image into two separate components - light-independent
+reflectance and structure-aware smooth illumination. The
+final estimated reflection component is treated as the en-
+hanced result.
+Unsupervised Learning: Although the above-mentioned
+methods perform well on synthetic data, they show limited
+generalization capability on real-world low-light images.
+This might be the result of overfitting. EnlightenGAN [20]
+proposed to solve this issue by adopting an unsupervised
+learning technique, i.e., avoiding the use of paired synthetic
+data. This work uses attention-guided UNet as a generator
+and global-local discriminators to achieve the objective of
+LLIE.
+Zero-short Learning: These methods, in low-level vision
+tasks, do not require any paired or unpaired training data.
+Zero-reference Deep Curve Estimation [13] formulates im-
+age enhancement as a task of image-specific deep curve
+estimation, taking into account pixel value range, mono-
+tonicity, and differentiability. It is a lightweight DCE-Net
+that doesn’t require paired or unpaired ground truth images
+during training and relies on non-reference loss functions
+that measure the enhancement quality hence driving the
+learning of the network. Another such method, Semantic-
+guided Zero-shot low-light enhancement Network [49] is a
+lightweight model for low-light enhancement factor extrac-
+tion which is inspired by the architecture of U-Net [37]. The
+output of this network is fed to a recurrent image enhance-
+ment network, along with the degraded input image. Each
+stage in this network considers the enhancement factor and
+the output from the previous scale as its input. This is fol-
+lowed by a feature-pyramid network that aims to preserve
+the semantic information in the image.
+More recently, researchers have experimented with trans-
+formers for Zero-shot Learning LLIE. Structure-Aware
+lightweight Transformer (STAR) [47] focuses on real-time
+image enhancement without using deep-stacked CNNs or
+large transformer models. STAR is formulated to capture
+long-range dependencies between separate image patches,
+facilitating the model to learn structural relationships be-
+tween different regions of the images. In STAR, patches of
+the image are tokenized into token embeddings. The tokens
+generated as an intermediate stage are passed to a long-
+short-range transformer that outputs two long and short-
+range structural maps. These structural maps can further
+predict curve estimation or transformation for image en-
+hancement tasks. Although these methods show impressive
+results for the study of low-light image enhancement for
+which it originally developed, they cannot deal with foggy
+low– light images.
+2.3. Limitations
+Previous works have relied on ASM-based methods in the
+case of dehazing and Retinex model-based methods for low-
+light image enhancement.
+However, these methods fail
+to generalize to real-world images. Recent deep learning-
+based methods using large networks solve the task of im-
+age dehazing and low-light enhancement separately. To our
+knowledge, no work is introduced that solves the two prob-
+lems in a collaborative network. Deep learning methods
+4
+
+also fail to generalize to different haze levels and darkness.
+3. Proposed Methodology
+In this section, we provide a detailed description of the over-
+all architecture proposed and the individual components in-
+cluded in the network.
+3.1. Architecture
+Like the group structure in [33], each group in our network
+consists of a K NAF Block [5] with a skip connection at
+the end as shown in Figure 3. The output of each group is
+concatenated, passed to the level attention module to find
+the weighted importance of the feature maps obtained, and
+post-processed using two convolutional layers. A long skip
+connection for global residual learning accompanies this.
+3.1.1
+NAF-Block
+To keep this work self-contained, we explain the NAF Block
+[5] in this subsection. NAF Block is the building block
+of Nonlinear Activation Free Network. Namely NAFNet
+[5]. To avoid over-complexity in the architecture, this block
+avoids using any activation functions like ReLU, GELU,
+Softmax, etc. hence keeping a check on the intra-block com-
+plexity of the network.
+The input first passes through Layer Normalization as it can
+help stabilize the training process. This is followed by con-
+volution operations and a Simple Gate (SG). SG is a variant
+of Gated Linear Units (GLU) [10] as evident from the fol-
+lowing equations 1 and 2
+GLU(X, f, g, σ) = f(X) ⊙ σ(g(X))
+(1)
+S impleGate(X, Y) = X ⊙ Y
+(2)
+and a replacement for GELU[18] activation function be-
+cause of the similarity between GLU and GELU (Equa-
+tion 3).
+GELU(x) = xφ(x)
+(3)
+In Simple Gate, the feature maps are divided into two parts
+along the channel dimension and then multiplied as shown
+in Figure 4. Another novelty introduced in this block is
+Simplified Channel Attention (SCA). Channel Attention
+(CA) can be expressed as:
+CA(X) = X ⊗ σ(W2max(0, W1pool(X)))
+(4)
+where X represents the feature map, pool indicates the
+global average pooling operation,σ is Sigmoid, W1, W2 are
+fully-connected layers and ⊗ is a channel-wise product op-
+eration. This can be taken as a special case of GLU from
+W
+W
+W
+H
+H
+H
+C/2
+C/2
+C/2
+Figure 4. Simple Gate as represented by Equation 2 ⊗ denotes
+channel-wise multiplicaWere
+which we can derivate the equation for Simplified Channel
+Attention:
+SCA(X) = X ⊗ Wpool(X)
+(5)
+3.1.2
+Level Attention Module
+Once we have extracted features from all the NAF Groups,
+we concatenate them and pass them through the Level At-
+tention Module (LAM) [45]. This module learns attention
+weights for features obtained at different levels.
+In LAM, each feature map is first reshaped to a 2D matrix
+of the size K × HWC, where K, H, W, and C are the no. of
+NAF Groups, height, width, and no. of channels of the fea-
+ture maps respectively. We find a correlation matrix of this
+2D matrix by multiplying it with its transpose matrix. Fi-
+nally, we multiply the 2D matrix with this correlation ma-
+trix and reshape it to K × H × W × C tensor. Inspired by
+residual learning, this tensor is substituted for residual and
+is added to the original concatenated feature maps. The re-
+sultant features are then reshaped to H × W × KC, passing
+through 1 × 1 convolution operation to get the H × W × C
+feature map. This is passed through some post-processing
+convolutions to get the final enhanced output. We include
+its architecture diagram in the supplementary material for a
+better understanding.
+3.2. Loss Functions
+Four loss functions, namely, reconstruction loss, perceptual
+loss, edge loss [19], and FFT loss[7], have been used to
+supervise the task of image restoration.
+The total loss L is defined in Equation 6, where λ1 = 0.04,
+λ2 = 1 and λ3 = 0.01.
+L = Ls + λ1Lp + λ2Le + λ3Lf
+(6)
+3.2.1
+Reconstruction Loss:
+The restored clear output image is compared with its ground
+truth value in the spatial domain using a standard l1 loss as
+5
+
+demonstrated in Equation 7. We use l1 loss instead of l2 loss
+as it does not over-penalize the errors and leads to better
+image restoration performance [48].
+Ls = 1
+N
+n
+�
+i=1
+∥ xgt
+i − NAFNet(xdark,hazy
+i
+) ∥1
+(7)
+In the above equation, xgt
+i refers to the ground truth clear im-
+age, and NAFNet(xdark,hazy
+i
+) denotes the output of our pro-
+posed NAFNet when a dark and hazy image is fed to the
+network.
+3.2.2
+Perceptual Loss:
+To reduce the perceptual loss and improve the image’s vi-
+sual quality, we utilize the features of the pre-trained VGG-
+19 network [38] obtained from the output of one of the
+ReLU activation layers. It is defined in Equation 8, where
+wi j, hij, and cij refer to the dimensions of the respective
+feature maps inside the VGG-19 architecture. φij denotes
+the feature maps outputted from the jth convolutional layer
+inside the i-th block in the VGG network.
+Lp =
+1
+wijhijci j
+wij
+�
+x=1
+hij
+�
+y=1
+cij
+�
+z=1
+∥ φij(Igt)xyz − φij(Iout)xyz ∥
+(8)
+3.2.3
+Edge Loss:
+To recover the high-frequency details lost because of the in-
+herent noise in dark and hazy images, we have an additional
+edge loss to constrain the high-frequency components be-
+tween the ground truth and the recovered image.
+Le =
+�
+(∇2(Igt) − ∇2(Iout))2 + ϵ2
+(9)
+In Equation 9, ∇2 refers to the Laplacian operation [22],
+which is then applied to the ground truth and the predicted
+clean image to get the edge loss.
+3.2.4
+FFT Loss:
+To supervise the haze-free results in the frequency domain,
+we add another loss called Fast Fourier transform (FFT) loss
+(denoted by Lf in Equation 12. It calculates the loss of both
+amplitude and phase using the l1 loss function without ad-
+ditional inference cost.
+Axgt
+i , Pxgt
+i = FFT(xgt
+i ),
+(10)
+Axout
+i , Pxout
+i
+= FFT(xout
+i ),
+(11)
+L f = 1
+N
+n
+�
+i=1
+(∥ Axgt
+i − Axout
+i
+∥1 + ∥ Pxgt
+i − Pxout
+i
+∥1)
+(12)
+4. Experimental Results
+To demonstrate the outcomes of our model’s approach to-
+wards image enhancement under low-visibility conditions,
+this section contains a detailed description of the dataset
+generated and used in Section 4.1, the experimental set-
+tings in Section 4.2, the metrics used for evaluation in Sec-
+tion 4.3 and a discussion on the results obtained in Sec-
+tion 5.1 and 5.2.
+4.1. Dataset Details
+Due to the lack of available datasets that meet our require-
+ments, we generate a new one using the AFO dataset [15].
+The dataset generation process has been elaborated below,
+and the final images have been shown in Figure 5.
+• Haze effect - To add fog, imgaug [21], a well-known
+python library was used. A random integral value be-
+tween 3, 4, 5 was selected, representing the fog’s sever-
+ity. For each image, this random number was chosen
+and pre-defined functions within the package were uti-
+lized to add a layer of fog to the image.
+• Low-light Effect - Given a normal image, our goal is
+to output a low-lit image while preserving the underly-
+ing information. We follow the pipeline introduced [8],
+which parametrically models the low light-degrading
+transformation by observing the image signal process-
+ing (ISP) pipeline between the sensor measurement
+system and the final image.
+The low-illumination-
+degrading pipeline is a three-stage process:
+– Unprocessing procedure - This part aims to syn-
+thesize RAW format images from input sRGB
+images by invert tone mapping, invert gamma
+correction, and the transformation of the image
+from sRGB space to cRGB space, and invert
+white balancing.
+– Low Light Corruption - This aims at adding shot
+and read noises to the output of the unprocess-
+ing procedure, as these are common in-camera
+imaging systems. Shot noise is a type of noise
+generated by the random arrival of photons in
+a camera, which is a fundamental limitation.
+Read noise occurs during the charge conversion
+of electrons into voltage in the output amplifier,
+which can be approximated using a Gaussian ran-
+dom variable with zero mean and fixed variance.
+– ISP Pipeline - RAW image processing is done
+after the lowlight corruption process in the fol-
+lowing order: add quantization noise, white bal-
+ancing from cRGB to sRGB, and gamma correc-
+tion, which finally outputs a degraded low-light
+image.
+6
+
+Ground Truth Images
+Ground Truth Images
+Generated Images
+Generated Images
+Figure 5. Visual illustration of a few sample images from our dataset. Columns 1 and 3 show original images taken from AFO Dataset
+[15], whereas Columns 2 and 4 show their corresponding images generated as explained in Section 4.1 simulating low-visibility conditions.
+• Combination of Haze and Low-light Effect - Re-
+sults of implementing the low-light generation algo-
+rithm described above on foggy images generated us-
+ing img-aug are shown here. It can be seen that com-
+bining the two (fog and low light) has introduced ad-
+versity in finding the location of the objects in the wa-
+ter bodies. Moreover, finding a unique solution for
+such a combination has not been explored to date
+4.2. Experimental Settings
+The images were resized to get the resultant dimensions as
+256 × 456. Adam optimizer with an initial learning rate of
+1e−4, β1, and β2 with a value of 0.9 and 0.999 were chosen.
+The batch size was fixed as 2. We have used 3 groups in all
+our experiments, each with 16 blocks. Pytorch backend was
+used to compile the model and train it.
+4.3. Evaluation Metrics
+We reported the results we obtained using two standard im-
+age restoration metrics (i.e., PSNR and SSIM). These met-
+rics will help us quantitatively evaluate the performance of
+our model in terms of feature colors and structure similarity.
+High PSNR and SSIM values if indicative of good results.
+5. Experimental Results
+The architecture used is given in Figure 2. This section
+gives a detailed analysis of the results obtained by the pro-
+posed method.
+5.1. Discussion and Comparison
+In this subsection, we discuss the evaluation results ob-
+tained by the proposed pipeline. Previous methods were
+Method
+Year
+PSNR
+SSIM
+Zero-DCE[13]
+2020
+12.323
+0.529
+SGZNet[49]
+2022
+12.578
+0.519
+BPPNet[39]
+2022
+15.507
+0.755
+DehazeNet[3]
+2016
+15.710
+0.391
+Star-DCE[47]
+2021
+16.651
+0.539
+FFANet[7]
+2020
+15.050
+0.582
+MSBDN-DFF[16]
+2020
+16.686
+0.689
+LVRNet (Ours)
+2022
+25.744
+0.905
+Table 1. Quantitative comparison of our proposed network
+with previous work. The best results and the second-best results
+have been highlighted with red color and blue colors, respectively.
+trained on the newly generated dataset and tested to com-
+pare their metrics with our model’s performance. These
+methods were built to enhance the low-light image or obtain
+a clear image from a hazy one. The results are mentioned
+in Table 1.
+We observe a huge increase in the PSNR value as compared
+to Zero-DCE[13], which enhances the low-light image as a
+curve estimation problem. However, it introduces an even
+amplified noise leading to color degradation as seen in Fig-
+ure 1. Notwithstanding its fast processing speed, Zero-DCE
+has limited noise suppression and haze removal capacity.
+Star-DCE[47], which uses a transformer backbone instead
+of a CNN one in the Zero-DCE network, shows a 35.12%
+increase in PSNR value. Owing to the added LAM struc-
+7
+
+S.no.
+Reconstruction Loss
+Perceptual Loss
+Edge Loss
+FFT Loss
+PSNR
+SSIM
+1.
+
+
+
+
+24.070
+0.870
+2.
+
+
+
+
+25.455
+0.903
+3.
+
+
+
+
+25.624
+0.897
+4.
+
+
+
+
+25.719
+0.900
+5.
+
+
+
+
+25.744
+0.905
+Table 2. Ablation experiments: We train our model using different combinations of loss functions to understand the importance of
+individual losses for image restoration. The best results are obtained when the model is trained using all the loss functions mentioned in
+this work.
+ture, using which our model can focus on more important
+feature maps, we can achieve a 54% higher PSNR value.
+SGZNet[49] uses pretrained networks for enhancement fac-
+tor estimation, thus their result is dependent on those pre-
+trained weights, leading to a lower PSNR value of 12.578
+on LowVis-AFO. From Figure 1, we observe that the result
+obtained from SGZNet is still degraded by excessive noise
+and lacks saturation. DehazeNet[3] is limited by the net-
+work’s depth and cannot generalize to real-world scenarios.
+Hence, it results in a low PSNR of 15.710. Methods like
+BPPNet[39] and FFANet[33] are end-to-end deep learning
+methods for image dehazing. BPPNet[39] distorts the color
+distribution in the recovered image as it cannot remove the
+dark regions, whereas FFA-Net[33] produces image with a
+lower perceptual quality.
+We propose an end-to-end deep learning pipeline (0.43M
+parameters) that can perform image dehazing and low-light
+image enhancement with a significant decrease in the num-
+ber of parameters as compared to MSBDN-DFF [16] (31M
+parameters) and FFA-Net[33] (4.45M parameters).
+The supplementary material has provided a discussion
+on the number of parameters of other models.
+We also
+trained the model for 10 epochs with fewer NAF blocks to
+prove that we achieved better results than the lighter results,
+not due to an increase in parameters but because of the self-
+sufficiency of the added LAM module, non-linear activation
+networks, and residual connections. The results of these ex-
+periments are reported in the supplementary material.
+5.2. Ablation Studies
+To prove the importance of the perceptual loss, edge loss,
+and fft-loss, added to supervise the training procedure, we
+conducted experiments excluding each of them and reported
+the values of PSNR and SSIM in Table 2. We keep the l1
+loss function constant in all experiments as it is critical in
+image restoration tasks. We observe an increase in metric
+values in the lower rows compared to row 1. As a result
+of more supervision in the unchanged architecture, there is
+an increase in the quality of clear images obtained, which
+are demonstrated in the supplementary material. There is
+also an increase in PSNR value (which depends on per-pixel
+distance) in row 3, once we train the model without percep-
+tual loss. This is seen as perceptual loss doesn’t compare
+individual pixel values but the high-level features obtained
+from a pretrained network. In row 4, we get a lower PSNR
+value on excluding edge loss compared to row 5, as we get
+lesser edge supervision. Overall, we get the best perfor-
+mance when we include all the loss functions, as seen in
+row 5.
+6. Conclusion
+In this work, we have presented Low-Visibility Restora-
+tion Network (LVRNet), a new lightweight deep learning
+architecture for image restoration.
+We also introduce a
+new dataset, LowVis-AFO, that includes a diverse combi-
+nation of synthetic darkness and haze. We also performed
+benchmarking experiments on our generated dataset and
+surpassed the results obtained using the previous image
+restoration network by a significant margin. Qualitative and
+quantitative comparison with previous work has demon-
+strated the effectiveness of LVRNet. We believe our work
+will motivate more research, focused on dealing with a com-
+bination of adverse effects such as haze, rain, snowfall, etc.
+rather than considering a single factor. In our future work,
+we plan to extend LVRNet for image restoration tasks where
+more factors, that negatively impact the image quality, are
+taken into account.
+8
+
+Supplementary Material
+To make our submission self-contained and given the page
+limitation, this supplementary material provides additional
+details. Section 1 gives an overview of the number of pa-
+rameters and PSNR obtained by different methods. Sec-
+tion 2 contains visual results that highlight the significance
+of the loss functions. Section 3 contains the ablation ex-
+periment with lesser blocks, and Section 4 demonstrates the
+architecture diagram of the level attention module.
+1. PSNR vs Parameters
+Figure 6 presents the PSNR vs. Parameters plot that the
+previous methods and our method achieved on the testing
+set of LowVis-AFO. Our model outperforms the state-of-
+the-art image dehazing and low-light image enhancement
+methods by a good margin while having a lesser number of
+parameters.
+Figure 6. The PSNR vs Number of Parameters of recent image
+restoration methods on the newly proposed LowVis-AFO dataset.
+S.no.
+#Blocks
+PSNR
+SSIM
+#params
+Runtime(s)
+1.
+14
+21.3432
+0.8626
+0.38M
+0.035
+2.
+12
+20.4302
+0.8488
+0.33M
+0.029
+3.
+10
+20.2965
+0.8494
+0.28M
+0.024
+Table 3. Results of the experiments conducted on a lesser num-
+ber of NAF blocks. The training was done for 10 epochs and the
+metrics were obtained on the test set thereafter.
+2. Ablation Experiment on Different Loss
+Functions
+Figure 8 demonstrates the visual results obtained when
+we conducted experiments excluding some loss functions.
+The motivation behind the experiment is to highlight the
+importance of the extra loss functions (perceptual loss, edge
+loss, fft-loss) added to supervise our pipeline. The quanti-
+tative results are given in Table 2 in the main manuscript.
+3. Ablation Experiment with Lesser Number of
+Blocks
+To prove the self-sufficiency of the individual components
+included in our architecture such as LAM, we conduct ex-
+periments with a lesser number of NAF blocks [5] and re-
+ported the PSNR and SSIM obtained in Table 1. Seeing
+the results, we can conclude that our model achieves better
+results, not because of an increase in the number of param-
+eters as compared to the lighter model, but because of the
+entire pipeline adopted.
+4. Level Attention Module
+As mentioned in the main text, the diagram for LAM[45]
+has been provided here in the supplementary material. (re-
+fer Figure 7)
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+26
+米
+Zero-DCE
+SGZNet
+24
+BPPNet
+DehazeNet
+22
+Star-DCE
+FFA-Net
+MSBDN-DFF
+PSNR
+20
+Ours
+18
+16
+14
+12
+0
+10M
+20M
+30M
+NumberofParametersStacked
+Feature Maps 
+K x H x W x C
+K x HWC
+Reshape
+Transpose
+Softmax
+K x HWC
+K x H x W x C
+Feature Maps
+Reshape
+H x W x KC
+Reshape
+H x W x C
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+LP
+LPF
+LPE
+LEF
+LPEF
+GT
+Input
+Figure 8. Qualitative results obtained from experiments conducted on different loss functions. In the figure, L = L1 loss, P = Perceptual
+Loss, E = Edge loss and F = FFT loss.
+12
+