Malaria detection using Deep Convolution Neural Network

Sumit Kumar*, Sneha Priya, Ayush Kumar

Georgia State University

§ ABSTRACT

The latest WHO report showed that the number of malaria cases climbed to 219 million last year, two million higher than last year. The global efforts to fight malaria have hit a plateau and the most significant underlying reason is international funding has declined. Malaria, which is spread to people through the bites of infected female mosquitoes, occurs in 91 countries but about 90% of the cases and deaths are in sub-Saharan Africa. The disease killed 4,35,000 people last year, the majority of them children under five in Africa. AI-backed technology has revolutionized malaria detection in some regions of Africa and the future impact of such work can be revolutionary. The malaria Cell Image Data-set is taken from the official NIH Website NIH data. The aim of the collection of the dataset was to reduce the burden for microscopists in resource-constrained regions and improve diagnostic accuracy using an AI-based algorithm to detect and segment the red blood cells. The goal of this work is to show that the state of the art accuracy can be obtained even by using $2$ layer convolution network and show a new baseline in Malaria detection efforts using AI.

§ INTRODUCTION

In the engineering , model-based design[1], model based control[2], and model based optimization[3] are the essential  components. The discovery of new data-driven models especially deep convolution neural networks in recent times that have worked well in image classification (AlexNet[4] ), engineering design[5], autonomous driving car[6, 7], radiology [8], human genome[9], and many more for developing state of the art[7] control and prediction systems are revolutionizing.
Convolution Neural Networks (CNN) are said to be inspired by biological processes that take place in the human brain and the connectivity pattern between neurons resembles the organization of the animal visual cortex. The convolutional layer is the core building block of a CNN. The layer's parameters consist of a set of learnable filters also called kernels, the parameters of the kernel are trained during the learning process and used during prediction. In 2012, a CNN called AlexNet[4] won the ImageNet Large Scale Visual Recognition Challenge. In later years, GoogLeNet[10] increased the mean average precision of object detection to 0.439329, and reduced classification error to 0.06656, the best result to date. The performance of GoogLeNet was close to or more than that of humans.
In this work, the goal is to use the CNN-based network in the classification of cell images to distinguish between parasitized and uninfected cell images. We found that a small network can work better than the current state of art network done so far on Malaria detection if the hyperparameter is tuned properly. The expected benefit of the small network is the deployment of a malaria detection system on resource-constrained devices like mobile phones and tablets where high-performance computing is not available in those regions.

§ THE PROBLEM

The most widely used method (so far) is examining thin blood smears under a microscope, and visually searching for infected cells. The patient’s blood is smeared on a glass slide and stained with contrasting agents to better identify infected parasites in their red blood cells. Then, a clinician manually counts the number of parasitic red blood cells, sometimes up to 5,000 cells (according to WHO protocol). Manually counting is error-prone and slow. A clinician takes 10 minutes to 30 minutes to count such a number as it is a time-consuming process. There are general guidelines that lab technicians should process no more than 25 slides each day, but a lack of qualified workers leads some to process four times as many.

Why a neural network?
Neural networks have performed really well in recent years in their ability to automatically extract features and learn filters and acted as a very good classifier of images. In previous machine learning solutions, features had to be manually programmed in, for example, size, color, and the morphology of the cells. Utilizing convolutional neural networks (CNN) will greatly speed up prediction time while mirroring (or even exceeding) the accuracy of clinicians.

Data is the basic building block of CNN-based predictor. Without data, the training of the network is not possible. Thankfully, we have a labeled and preprocessed dataset of cell images to train and evaluate our model. NIH[11] has a malaria dataset of 27,558 cell images with an equal number of parasitized and uninfected cells. Datasets contains 2 folders:$1$ Infected ; $2$ Uninfected.
This total $27,558$ images are of variable size color images with equal instances of parasitized and uninfected cells from the thin blood smear slide images from the Malaria Screener research activity. Giemsa-stained thin blood smear slides from 150 patients, falciparum-infected and 50 healthy patients were collected and photographed at Chittagong Medical College Hospital, Bangladesh. The smartphone’s built-in camera acquired images of slides for each microscopic field of view. The images were manually annotated by an expert slide reader at the Mahidol-Oxford Tropical Medicine Research Unit in Bangkok, Thailand.

§ APPROACH

Classification of cell images is an interesting problem and has great utility. There are already country and medical universities which are leveraging this AI-backed technology that can detect diseases(refer to related works). I choose LeNet5[12] as the starting model, which can work with grayscale images. The Le-Net5 network is modified for multi-channel images (as images are on RGB scale) to work with 3 layers of images. and further hyperparameter tuning[13] is done to classify colored cell images with an efficiency target of 95% or more on the test data set.

A deep convolution neural network uses an algorithm with millions of pictures as input to train before it is able to generalize the input and make predictions for images it has never seen before.
To teach an algorithm how to recognize objects in images, we use a specific type of Artificial Neural Network: a Convolutional Neural Network (CNN). Their name stems from one of the most important operations in the network: convolution.
Convolutional Neural Networks are inspired by the brain. Research in the 1950s and 1960s by D.H Hubel and T.N Wiesel on the brain of mammals suggested a new model for how mammals perceive the world visually. They showed that cat and monkey visual cortexes include neurons that exclusively respond to neurons in their direct environment.
The computer world consists of only numbers. Every image can be represented as multi-dimensional arrays of numbers, known as pixels.
CNNs, like neural networks, are made up of neurons with learnable weights and biases. Each neuron receives several inputs, takes a weighted sum over them, passes it through an activation function, and responds with an output. The whole network has a loss function.
Unlike feed-forward neural networks, where the input is a vector, where the input is a multi-channeled image (3 channel in this case).
The convolution layer is the main building block of a convolutional neural network.
The convolution layer comprises a set of independent filters. Each filter is independently convolved with the image and we end up with multi-layer feature maps.
All these filters are initialized randomly and become our parameters which will be learned by the network subsequently.
Parameter sharing is sharing of weights by all neurons in a particular feature map.
Local connectivity is the concept of each neural connected only to a subset of the input image (unlike a neural network where all the neurons are fully connected)
This helps to reduce the number of parameters in the whole system and makes the computation more efficient.
A pooling layer is another building block of a CNN. Its function is to progressively reduce the spatial size of the representation to reduce the number of parameters and computations in the network. The pooling layer operates on each feature map independently.
Batch normalization is a method we can use to normalize the inputs of each layer, in order to fight the internal covariate shift problem.
During training time, a batch normalization layer does the following: it first  calculate the mean and variance of the layer's input, then it normalizes the layer inputs using the previously calculated batch statistics, and last, it scales and shifts in order to obtain the output of the layer.
Dropout is a regularization technique for neural network models. Dropout technique randomly selected neurons are ignored during training. They are “dropped out” randomly. This means that their contribution to the activation of downstream neurons is temporally removed on the forward pass and any weight updates are not applied to the neuron on the backward pass.

Data: The data set was downloaded from the NIH website[11]. A sample of image for both 'not infected cell' and 'infected cell' is shown below. On the left, we have a cell image, which is not infected by the malaria parasite. The infected cell image contains violet dots, which represent the malaria plasmodium parasite during imaging.

Input data image of cell (left : Not infected cell) ; (right : Infected cell)

After hyperparameter search, our final architecture consists of two convolution layers.  The input data is preprocessed and reduced its image size to $64 \times 64$, which is fed into a convolution layer with 32 filters of $3 \times 3$ size, with a stride of $1$. The result of convolution is $62 \times 62 \times 32$ feature array. Then the feature array is fed to a max-pooling layer of size $2 \times 2$ and the resultant feature matrix becomes a $31x31x32$. Max pooling[14] reduces the computational resources required by reducing the size of the pixel of the feature matrix while keeping the feature intact. Regularisation[15] techniques called batch normalization is deployed across all $32$ parallel channels of the output of the max-pooling layer. The batch normalized feature matrix is added with a dropout layer, with a dropout factor equal to $0.2$. The convolution layer, max pooling layer, batch normalization layer, and drop-out[16] complete one layer convolution process in the architecture.  The architecture is shown below.

The architecture of CNN

The output of one complete convolution layer is given to another convolution layer with 32 filters of $3 \times3$ size, with a stride of 1. The result of convolution is $29\times 29 \times 32$ feature array. Then the feature array is fed to max-pooling layer of size $2\times2$ and the resultant feature matrix becomes a $14\times14\times32$. Then, batch normalization is deployed across all $32$ parallel channels of the output of the max-pooling layer. The batch normalized feature matrix is added with a dropout layer, with a dropout factor equal to $0.2$.

After the two complete convolution layers, the feature size has reduced enough and most of the feature is extracted so that we can now connect it with the feed-forward network. The $14\times14\times32$ feature matrix is flattened which size becomes 6272. This flattened feature is fed to a feed-forward network of size 512, along with batch normalization and dropout. The feed-forward network is further added to another layer of 256 neurons with batch normalization and dropout. Finally, the output layer is connected with 2 neurons, and the activation function in the output layer is softmax. The activation of the rest of the layers is 'Rectified Linear unit(RELU)[17]'. The cost function for the error measurement is used as categorial_crossentropy, and the optimizer is 'adam'.
The number of parameters to be learned are 3,357,090 (approx 3.5 million). With the available computation capacity, it took 20 minutes for training. A summary of the model is shown below.

Summary of Neural Network model

The optimiser that is used is called Adaptive Moment Estimation (Adam) [18],which combines ideas from both RMSProp and Momentum. It computes adaptive learning rates for each parameter.

Algorithm used during training

The loss function deployed is 'categorical_crossentropy'.Cross-entropy loss, or log loss, measures the performance of a classification model whose output is a probability value between 0 and 1. Cross-entropy loss increases as the predicted probability diverge from the actual label. So predicting a probability of .012 when the actual observation label is 1 would be bad and result in a high loss value. A perfect model would have a log loss of 0.

§ RESULTS

The model is trained and the result of the model and the cost function is shown below.
The training and validation error is calculated simultaneously as the training progress. The number of epochs is set to 50, while a callback function is written to invoke early stopping when the model's validation accuracy reaches 95% of accuracy. This is done to avoid overfitting (generalization error). If a callback is not deployed, the model over-fits and memorizes the training data. In such circumstances, the model gives 98% accuracy on training data but performs badly on validation and testing data.  Once the training is done (which stops when validation accuracy reached 95%), then the testing error is calculated on the trained model. The testing accuracy achieved around 95.4%, which is satisfactory.

Result of training and testing

The plot of the cost function is calculated and it is plotted with respect to batch progression. The cost function initially drops sharply and then the rate is reduced. The batch gradient descent is used with a batch size of 64 images per batch. The result of batch training is the cost becomes noisy and fluctuates, but the overall trend of progression is toward optimal value. The cost function is plotted below.

Plot of cost with respect to batch

§ RELATED WORK

Machine learning is being used in control[19, 20, 21, 22, 23, 24] to model[25, 26] and prediction of complex systems [27, 28, 29, 30, 31, 32, 33].

There are various works in using machine learning in Malaria detection [34, 35] used deep CNN for Malaria detection but their model is more complex than ours. [34] used stacking-based approach for automated quantitative detection of Plasmodium falciparum malaria from blood smear. For the detection, a custom designed convolutional neural network (CNN) operating on focus stack of images is used. The cell counting problem is addressed as the segmentation problem and we propose a 2-level segmentation strategy. Use of CNN operating on focus stack for the detection of malaria improved the detection accuracy (both in terms of sensitivity [97.06%] and specificity [98.50%]) but also favored the processing on cell patches and avoided the need for hand-engineered features. [36] used machine learning for detection of Malaria.
In 2016, Uganda's Ministry of Health found that the disease is the leading cause of death in the country - accounting for 27 per cent of deaths.
Mortality rates are particularly high in rural areas, where the lack of doctors and nurses is acute. Nursing assistants are often taught to read slides instead, but inadequate training can lead to misdiagnosis. Due to lack of availability of lab technicians in the region lead some to process four times as many as recommended number of screening in a day, while it is recommended that each technician should process no more than 25 slides each day.
There are so many patients who may require malaria and TB tests, and overworking day and night. The AI lab([37], at Makerere University, has developed a way to diagnose the blood samples using a cell phone.
The program learns to create its own criteria based on a set of images that have been presented to it previously. It learns to recognize the common features of the infections. The smartphone clamped in place over one microscope eyepiece brings to light a detailed image of the blood sample below - each malaria parasite circled in red by artificially intelligent software.
With this AI backed technology, pathogens are counted and mapped out quickly, ready to be confirmed by a health worker. Diagnosis times could be slashed from 30 minutes to as little as two minutes.
The AI software is built on deep learning algorithms that use an annotated library of microscope images to learn the common features of plasmodium parasites that cause malaria. Along with malaria paracites this lab also diagnosis the bacteria called Mycobacterium tuberculosis that is responsible for tuberculosis.
Researchers at the Lister Hill National Center for Biomedical Communications (LHNCBC), part of National Library of Medicine (NLM), have developed a mobile application that runs on a standard Android smartphone attached to a conventional light microscope.

§ CONCLUSIONS

In this work , we showed a state of the art $2$ layer deep convolution network for Malaria diagnosis and detection. This method works with state of the art level accuracy without any expensive preprocessing.

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SNSN-323-63 March 30, 2021

Measuring standard model parameters using top-quark
cross sections in ATLAS and CMS

Matteo M. Defranchis
on behalf of the ATLAS and CMS Collaborations

CERN, Geneva, Switzerland

> Theoretical predictions of top-quark cross sections depend on the values of
> the parameters of the quantum chromodynamics Lagrangian, such as the strong
> coupling constant and the mass of the top quark, but also on parameters of
> the electroweak sector of the standard model and parton distribution
> functions. Comparisons between state-of-the-art calculations and recent
> measurements at the ATLAS and CMS experiments at the CERN LHC allow for the
> precise determination of these parameters. In this proceeding, the most
> recent results by the two collaborations are summarized and discussed.

> PRESENTED AT
>
>
>
>
> $13^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Durham, UK (videoconference), 14–18 September, 2020

##  1 Introduction

In proton-proton (pp) collisions at the CERN LHC, top quarks are mainly
produced in quark-antiquark pairs ($\mathrm{t\overline{t}}$) via the mechanism
of gluon fusion. The $\mathrm{t\overline{t}}$ production cross section can be
calculated in perturbative quantum chromodynamics (QCD), and is currently know
up to the next-to-next-to-leading order (NNLO) in perturbation theory,
including next-to-next-to-leading logarithmic (NNLL) corrections [3].
Differential and multi-differential calculations at NNLO are also becoming
available for an increasing number of variables [4]. The predicted cross
sections depend on the values of fundamental parameters of the QCD Lagrangian
such as the mass of the top quark, $m_{\mathrm{t}}$, and the strong coupling
constant, $\alpha_{\mathrm{S}}$, but also on the parton distribution functions
(PDFs) of the proton. Therefore, measurements of the inclusive and
differential $\mathrm{t\overline{t}}$ production cross sections can be used to
extract the values of $m_{\mathrm{t}}$, $\alpha_{\mathrm{S}}$, and to probe
the proton PDFs. In addition, this method allows the value of $m_{\mathrm{t}}$
to be determined in a well-defined renormalization scheme (on-shell,
$\mathrm{\overline{MS}}$), overcoming the difficulties related to the
interpretation of the results of direct $m_{\mathrm{t}}$ measurements. These
are often referred to as top quark Monte Carlo (MC) mass
($m_{\mathrm{t}}^{\mathrm{MC}}$) measurements, as the results depends on the
details of the MC simulation [5]. However, it has to be considered that cross
section measurements have a residual dependence on the value of
$m_{\mathrm{t}}^{\mathrm{MC}}$ due to acceptance corrections, which has to be
taken into account. Furthermore, the value of $m_{\mathrm{t}}^{\mathrm{MC}}$
also affects the efficiency of most kinematic reconstruction methods, which
are often used in differential cross section measurements.

In this proceeding, the most recent results obtained by the ATLAS [1] and CMS
[2] Collaborations are summarized and discussed. The results are obtained
using pp collision data at different centre-of-mass energies. In certain
cases, the precision of the measurements outperform the one of the
corresponding fixed-order calculations, making theoretical uncertainties a
limiting factor.

##  2 Extraction of QCD parameters

The most precise measurement to date of the inclusive $\mathrm{t\overline{t}}$
production cross section, $\sigma_{\mathrm{t\overline{t}}}$, has been obtained
by the ATLAS Collaboration using 36.1$\;\mathrm{fb}^{-1}$ of pp collisions
data at $\sqrt{s}=13\,\mathrm{TeV}$ [6]. The measurement is performed with
$\mathrm{t\overline{t}}$ candidate events in the $\mathrm{e^{\mp}}\mu^{\pm}$
channel, and benefits from significant improvements in the calibration of the
lepton identification efficiencies and the integrated luminosity. The cross
section is measured to be $\sigma_{\mathrm{t\overline{t}}}=826.4\pm
3.6~{}(\mathrm{stat})\pm 19.6~{}(\mathrm{syst})~{}\mathrm{pb}$, corresponding
to a total uncertainty of 2.4%. The result is in good agreement with the
theoretical prediction at NNLO+NNLL
$\sigma_{\mathrm{t\overline{t}}}^{\mathrm{th}}=832~{}^{+20}_{-29}~{}\mathrm{(scale)}\pm
35~{}(\mathrm{PDF+\alpha_{\mathrm{S}}})~{}\mathrm{pb}$ [3], that has a total
uncertainty of about 5%. The measured $\sigma_{\mathrm{t\overline{t}}}$ is
used to extract the value of the top quark pole mass
($m_{\mathrm{t}}^{\mathrm{pole}}$) by comparing to the NNLO+NNLL prediction
(left-hand plot in Figure 1), resulting in
$m_{\mathrm{t}}^{\mathrm{pole}}=173.1~{}^{+2.0}_{-2.1}\,\mathrm{GeV}$. Here,
the precision of the result is limited by QCD scale and PDF uncertainties. In
the procedure, the dependence of the measured cross section on
$m_{\mathrm{t}}^{\mathrm{MC}}$ is taken into account by assuming
$m_{\mathrm{t}}^{\mathrm{pole}}=m_{\mathrm{t}}^{\mathrm{MC}}$, as shown in
Figure 1 (left). The result for $\sigma_{\mathrm{t\overline{t}}}$ has also
been used to estimate cross section ratios at different centre-of-mass
energies, which can be used to probe the validity of existing PDF sets.

Figure 1: Left: Measured (markers) and predicted (band)
$\sigma_{\mathrm{t\overline{t}}}$ at $\sqrt{s}=13\,\mathrm{TeV}$ as a function
of $m_{\mathrm{t}}^{\mathrm{pole}}$, assuming
$m_{\mathrm{t}}^{\mathrm{MC}}=m_{\mathrm{t}}^{\mathrm{pole}}$. These results
are used to extract the value of $m_{\mathrm{t}}^{\mathrm{pole}}$ from the
measured $\sigma_{\mathrm{t\overline{t}}}$ [6]. Right: Running of the top
quark mass as a function of the scale $\mu_{m}$, defined as half of the
invariant mass of the $\mathrm{t\overline{t}}$ system, using theoretical
calculations with fixed or dynamic scales [12].

A different approach is followed by the CMS Collaboration in a measurement
where $\sigma_{\mathrm{t\overline{t}}}$ is measured simultaneously with
$m_{\mathrm{t}}^{\mathrm{MC}}$ via a profile maximum-likelihood fit to multi-
differential distributions [7]. The analysis is performed with
35.9$\;\mathrm{fb}^{-1}$ of pp collision data at $\sqrt{s}=13\,\mathrm{TeV}$,
using $\mathrm{t\overline{t}}$ candidate events in the
$\mathrm{e^{\mp}}\mu^{\pm}$ channel. The cross section is measured to be
$\sigma_{\mathrm{t\overline{t}}}=815\pm 2~{}(\mathrm{stat})\pm
35~{}(\mathrm{syst})~{}\mathrm{pb}$, in good agreement with the NNLO+NNLL
theoretical prediction and the ATLAS result. The measured
$\sigma_{\mathrm{t\overline{t}}}$ is used to separately extract the value of
$\alpha_{\mathrm{S}}$ and $m_{\mathrm{t}}$ by comparing to NNLO predictions in
the $\mathrm{\overline{MS}}$ scheme, using different sets of PDFs. The values
of $m_{\mathrm{t}}$ and $\alpha_{\mathrm{S}}$ are determined with a precision
of 1.2% and about 2%, respectively. However, it has to be noted that the
values of $m_{\mathrm{t}}$ and $\alpha_{\mathrm{S}}$ cannot be determined
simultaneously from a measurement of $\sigma_{\mathrm{t\overline{t}}}$.

A subsequent analysis by the CMS Collaboration was designed to determine
simultaneously the values of $\alpha_{\mathrm{S}}$, $m_{\mathrm{t}}$, and the
PDFs from a triple-differential measurement of
$\sigma_{\mathrm{t\overline{t}}}$. The analysis is performed with
35.9$\;\mathrm{fb}^{-1}$ of collisions data at $\sqrt{s}=13\,\mathrm{TeV}$,
using $\mathrm{t\overline{t}}$ candidate events in the dileptonic channels
[8]. In this measurement, the invariant mass of the $\mathrm{t\overline{t}}$
system is chosen for is sensitivity to $m_{\mathrm{t}}$ near the production
threshold, the number of jets for its sensitivity to the value of
$\alpha_{\mathrm{S}}$, while the rapidity of the $\mathrm{t\overline{t}}$
system carries additional information on the PDFs. The sensitivity of the
analysis on the assumption of $m_{\mathrm{t}}^{\mathrm{MC}}$ is mitigated by
means of a dedicated kinematic reconstruction algorithm that does not depend
on any assumption on the value of $m_{\mathrm{t}}$. A QCD analysis is then
performed at NLO by combining the measured multi-differential cross section
with HERA deep inelastic scattering data. The inclusion of
$\mathrm{t\overline{t}}$ data allows for a significant reduction of the
uncertainty in the gluon PDF at momentum fractions on the order of
$x_{\mathrm{B}}=0.1$. Moreover, this method allows the correlation between the
gluon PDF and the value of $\alpha_{\mathrm{S}}$ to be significantly reduced
compared to standard PDF determinations. The values of $\alpha_{\mathrm{S}}$
and $m_{\mathrm{t}}$ are also determined with remarkable precision: The top
quark pole mass is measured to be $m_{\mathrm{t}}^{\mathrm{pole}}=170.5\pm
0.8\,\mathrm{GeV}$, the most precise result, to date. It is however important
to note that threshold effects that are not considered in this analysis could
have a significant impact on the result, as shown by a dedicated study of
next-to-leading power (NLP) Coulomb corrections [9].

Similarly, the ATLAS Collaboration has performed a PDF global fit at NNLO in
QCD with NLO electroweak corrections, including HERA data in combination with
W and Z cross section measurements at $\sqrt{s}=7\,\mathrm{TeV}$ and
$\mathrm{t\overline{t}}$ observables measured at $\sqrt{s}=8\,\mathrm{TeV}$.
Similarly to the CMS analysis, a reduction of the PDF uncertainty at high
momentum fraction is achieved [10].

The sensitivity of the $\mathrm{t\overline{t}}$ production threshold to
$m_{\mathrm{t}}$ is exploited by the ATLAS Collaboration in a measurement
where the top quark mass is determined at NLO in the pole and
$\mathrm{\overline{MS}}$ schemes. The analysis is performed with
20.2$\;\mathrm{fb}^{-1}$ of pp collisions data at $\sqrt{s}=8\,\mathrm{TeV}$,
using $\mathrm{t\overline{t}}$ candidate events in the lepton+jets final state
[11]. Here, the sensitivity is further increased by considering the invariant
mass of the $\mathrm{t\overline{t}}$ $+1$ jet system, resulting in
$m_{\mathrm{t}}^{\mathrm{pole}}=171.1\pm
1.0~{}\mathrm{(exp)}~{}^{+0.7}_{-0.3}~{}\mathrm{(th)}\,\mathrm{GeV}$ and
$m_{\mathrm{t}}(m_{\mathrm{t}})=162.9\pm
1.1~{}\mathrm{(exp)}~{}^{+2.1}_{-1.2}~{}\mathrm{(th)}\,\mathrm{GeV}$ for the
top-quark pole and $\mathrm{\overline{MS}}$ masses, respectively. The larger
theoretical uncertainty in the value of $m_{\mathrm{t}}(m_{\mathrm{t}})$
reflects the increased dependence on the choice of QCD scales in the
$\mathrm{\overline{MS}}$ scheme near the production threshold.

In the $\mathrm{\overline{MS}}$ scheme, the value of the top quark mass
$m_{\mathrm{t}}(\mu_{m})$ depends on the scale $\mu_{m}$ at which it is
evaluated. Similarly to $\alpha_{\mathrm{S}}$, this is effect is commonly
referred to as the running of $m_{\mathrm{t}}$, and can be probed by
extracting the value of $m_{\mathrm{t}}(\mu_{m})$ at different energy scales.
This has been investigated for the first time by the CMS Collaboration, where
the running of $m_{\mathrm{t}}$ is extracted from a differential measurement
of $\sigma_{\mathrm{t\overline{t}}}$ as a function of the invariant mass of
the $\mathrm{t\overline{t}}$ system [12]. The analysis is performed using
35.9$\;\mathrm{fb}^{-1}$ of pp collisions data at $\sqrt{s}=13\,\mathrm{TeV}$
with $\mathrm{t\overline{t}}$ candidate events in the
$\mathrm{e^{\mp}}\mu^{\pm}$ channel. The cross section is unfolded to the
parton level by means of a maximum-likelihood method, in which the dependence
on $m_{\mathrm{t}}^{\mathrm{MC}}$ is mitigated and taken into account. The
extraction of the running is then performed using NLO calculations where the
value of $\mu_{m}$ is either kept constant or set to the relevant energy
scale, following the method discussed in Ref. [13]. The results are shown in
Figure 1 (right). In both cases a hypothetical no-running scenario is excluded
at above 95% confidence level. The result can be further improved by making
use of the newly available predictions at NNLO in the $\mathrm{\overline{MS}}$
scheme [13].

##  3 Extraction of electroweak parameters

Electroweak contributions to the process of $\mathrm{t\overline{t}}$
production, such as the exchange of a massive neutral boson between the two
final-state top quarks, introduce a dependence of the $\mathrm{t\overline{t}}$
production cross section on fundamental electroweak parameters such as the
Yukawa coupling between the top quark and the Higgs boson. The impact of
electroweak corrections is particularly relevant in the proximity of the
$\mathrm{t\overline{t}}$ production threshold. This is exploited by a recent
measurement by the CMS Collaboration, performed with 137$\;\mathrm{fb}^{-1}$
of pp collisions data at $\sqrt{s}=13\,\mathrm{TeV}$, where top-Higgs Yukawa
coupling is extracted from a measurement of multi-differential distributions
of dileptonic $\mathrm{t\overline{t}}$ events [14]. Proxy variables for the
invariant mass of the $\mathrm{t\overline{t}}$ system and the rapidity
difference between the top and antitop quark are built using reconstructed
leptons and b-tagged jets, in order to avoid any possible bias introduced by
the kinematic reconstruction of the $\mathrm{t\overline{t}}$ system. The
Yukawa coupling is then extracted by comparing to NLO MC predictions to which
electroweak corrections are applied. The ratio between the measured Yukawa
coupling and the standard model expectation is found to be
$1.16~{}^{+0.24}_{-0.35}$, achieving comparable sensitivity to direct
measurements with on-shell Higgs bosons.

Finally, the production cross section of a single top quark in the $t$-channel
has been used by the CMS Collaboration to perform the first direct and model-
independent measurement of elements of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix. The analysis, performed using 35.9$\;\mathrm{fb}^{-1}$ of pp
collisions data at $\sqrt{s}=13\,\mathrm{TeV}$, exploits multivariate
techniques in order to increase the sensitivity to the values of
$|V_{\mathrm{tb}}|$ and $|V_{\mathrm{td}}|^{2}+|V_{\mathrm{ts}}|^{2}$ [15].
Different scenarios of physics beyond the standard model are considered, and
the total decay width of the top quark is also constrained. In this analysis,
the precision in the value of $|V_{\mathrm{tb}}|$ is improved by about 50%
compared to a previous similar measurement by the CMS Collaboration.

##  4 Summary

Measurements of top-quark cross sections can be used to extract parameters of
the QCD Lagrangian, such as $\alpha_{\mathrm{S}}$ and $m_{\mathrm{t}}$, but
also of the electroweak sector, such as the top-Higgs Yukawa coupling and
elements of the CKM matrix. The same measurements can also be used to probe
the PDFs of the proton at large momentum fractions. In this proceeding, the
most recent results by the ATLAS and CMS Collaborations have been summarized
and discussed. Significant progress has been made with the use of LHC Run2
data at $\sqrt{s}=13\,\mathrm{TeV}$, and the increasing availability of
theoretical predictions at higher orders in perturbation theory offers
prospects of significant improvements in the near future.

## References

  * [1] ATLAS Collaboration, JINST 3 (2008) S08003
  * [2] CMS Collaboration, JINST 3 (2008) S08004
  * [3] M. Czakon and A. Mitov, Comput. Phys. Commun. 185 (2014) 2930
  * [4] S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli, JHEP 07 (2019) 100
  * [5] M. Butenschoen, B. Dehnadi, A. H. Hoang, V. Mateu, M. Preisser and I. W. Stewart, Phys. Rev. Lett. 117 (2016) 232001
  * [6] ATLAS Collaboration, Eur. Phys. J. C 80 (2020) 528
  * [7] CMS Collaboration, Eur. Phys. J. C 79 (2019) 368
  * [8] CMS Collaboration, Eur. Phys. J. C 80 (2020) 658
  * [9] W. L. Ju, G. Wang, X. Wang, X. Xu, Y. Xu and L. L. Yang, JHEP 06 (2020) 158
  * [10] ATLAS Collaboration, ATL-PHYS-PUB-2018-017
  * [11] ATLAS Collaboration, JHEP 11 (2019) 150
  * [12] CMS Collaboration, Phys. Lett. B 803 (2020) 135263
  * [13] S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli, JHEP 08 (2020) 027
  * [14] CMS Collaboration, Phys. Rev. D 102 (2020) 092013
  * [15] CMS Collaboration, Phys. Lett. B 808 (2020) 135609


# Translation-like actions yield regular maps

Yongle Jiang  Department of Mathematics, SUNY at Buffalo, USA, 14260–2900
<EMAIL_ADDRESS>
###### Abstract.

For finitely generated groups $H$ and $G$, we observe that $H$ admits a
translation-like action on $G$ implies there is a regular map, which was
introduced in Benjamini, Schramm and Timár’s joint paper, from $H$ to $G$.
Combining with several known obstructions to the existence of regular maps, we
have various applications. For example, we show that the Baumslag-Solitar
groups do not admit translation-like actions on the classical lamplighter
group.

##  1\. Introduction

The concept of translation-like actions was introduced by Whyte to solve a
geometric version of the von Neumann conjecture [whyte]. It serves as a
geometric analogy of subgroup containment for finitely generated groups: if
$H$ is a subgroup of a finitely generated group $G$, then the natural right
action of $H$ on $G$ is a translation-like action. Later on, answering a
question of Whyte, Seward solved a geometric version of Burnside’s problem
which is formulated using translation-like actions [seward]. Recently,
translation-like actions play an important role in studying the question which
finitely generated groups admit a weakly aperiodic shift of finite type
[jeandel]. Similar approaches (using notions from geometric group theory) to
this question also appeared in [cohen_adv., cp].

Despite the above success of application of this concept in various problems,
it seems to us that not too much is known on whether a group $H$ admits a
translation-like action on another finitely generated group $G$ assuming $H$
is not a subgroup of $G$. For known examples, see [cohen, jeandel, seward,
whyte]. The purpose of this note is to observe that translation-like actions
yield regular maps, which was introduced in [separation].

######  Theorem 1.1 (Main theorem).

Let $G$ and $H$ be finitely generated groups. If $H$ admits a translation-like
action on $G$, then $H\rightarrow_{reg}G$.

It is well-known that there are several ways to rule out existence of regular
maps between spaces: asymptotic dimension, Dirichlet harmonic functions,
growth and separation (see [separation, last paragraph in section 1]). By the
above theorem, we also get obstructions to admitting translation-like actions
for a pair of groups. We discuss these in Section 3.

##  2\. Definitions and proof of the main theorem

First, we recall the general definition of translation-like actions following
Seward [seward, Definition 1.1], but here we use left actions. Note that the
original definition is due to Whyte [whyte, Definition 6.1].

######  Definition 2.1 (Translation-like actions).

Let $H$ be a group and let $(X,d)$ be a metric space. A left action $*$ of $H$
on $X$ is _translation-like_ if it satisfies the following two conditions:

  * (i)

The action is free (i.e. $h*x=x$ implies $h=1_{H}$).

  * (ii)

For every $h\in H$ the set $\\{d(x,h*x)|~{}x\in X\\}$ is bounded.

Next, we recall the definition of regular maps following [separation].

######  Definition 2.2 (Regular maps).

Let $(Z,d_{Z}),(X,d_{X})$ be metric spaces, and let $\kappa<\infty$. A (not
necessarily continuous) map $f:Z\to X$ is _$\kappa$ -regular_ if the following
two conditions are satisfied.

  * (i)

$d_{X}(f(z_{0}),f(z_{1}))\leq\kappa(1+d_{Z}(z_{0},z_{1}))$ holds for every
$z_{0},z_{1}\in Z$;

  * (ii)

For every open ball $B=B(x_{0},1)$ with radius 1 in $X$, the inverse image
$f^{-1}(B)$ can be covered by $\kappa$ open balls of radius 1 in $Z$.

A _regular_ map is a map which is $\kappa$-regular for some finite $\kappa$.
Write $Z\rightarrow_{reg}X$ if there is a regular map from $Z$ to $X$.

Note that as mentioned in [separation, p. 5], if there is a quasi-isometry
between bounded degree graphs $Z$ and $X$, then there is a regular map from
$Z$ to $X$ and also the other direction. Hence being quasi-isometric implies
$Z\rightarrow_{reg}X$ and $X\rightarrow_{reg}Z$, but the converse does not
hold.

From now on, we fix a finitely generated group $H$ and take $(X,d)$ to be a
finitely generated group $G$ using a left-invariant word length metric $d$
associated to some finite symmetric generating set $T$, i.e.
$d_{G}(x,y):=\ell_{G}(x^{-1}y)$ for all $x,y\in G$, where $\ell_{G}$ is the
word length associated to $T$.

If $H\overset{*}{\curvearrowright}G$ is a translation-like action, then we can
define a map $L:H\times G\to G$ by setting $L(h,x)=x^{-1}(h*x)$ for all $x\in
G,h\in H$.

Note that $\overset{*}{\curvearrowright}$ is an action implies that
$L(h_{1},x)L(h_{2},h_{1}*x)=L(h_{2}h_{1},x)$ for all $h_{1},h_{2}\in H$ and
$x\in G$.

Write $c(g,x):=L(g,x)^{-1}$, then $c:H\times G\to G$ is a cocycle in the usual
sense, i.e. $c(h_{1}h_{2},x)=c(h_{1},h_{2}*x)c(h_{2},x)$ for all
$h_{1},h_{2}\in H,x\in G$. And the two conditions in the definition of
translation-like action are just the following.

  1. (1)

For all $x\in G$, the map $H\ni h\mapsto c(h,x)\in G$ is 1-1.

  2. (2)

For all $h\in H$, the set $\\{c(h,x):x\in G\\}$ is bounded, i.e. $\sup_{x\in
G}\ell_{G}(c(h,x))=:\lambda_{h}<\infty$.

We are ready to prove our main theorem.

###### Proof of Theorem 1.1.

Fix any $x\in G$, define $f:H\to G$ by $f(h):=c(h^{-1},x)^{-1}$. We claim this
map is a regular map.

Fix a symmetric generating set $S$ for $H$ and define $\kappa:=\max_{s\in
S}\lambda_{s}+\\#T$, where $\lambda_{h}:=\sup_{x\in G}\ell_{G}(c(h,x))$ for
all $h\in H$.

Note that the second condition in Definition 2.2 is clear since $h\mapsto
c(h,x)$ is 1-1. We are left to check the first condition holds.

$\displaystyle d_{G}(f(h_{1}),f(h_{2}))$ $\displaystyle=$ $\displaystyle
d_{G}(c(h_{1}^{-1},x)^{-1},c(h_{2}^{-1},x)^{-1})$ $\displaystyle=$
$\displaystyle\ell_{G}(c(h_{1}^{-1},x)c(h_{2}^{-1},x)^{-1})$ $\displaystyle=$
$\displaystyle\ell_{G}(c(h_{1}^{-1},x)c(h_{2},h_{2}^{-1}*x))$ $\displaystyle=$
$\displaystyle\ell_{G}(c(h_{1}^{-1}h_{2},h_{2}^{-1}*x))$ $\displaystyle=$
$\displaystyle\ell_{G}(c(s_{1}\cdots s_{k},h_{2}^{-1}*x)),\mbox{~{}write
$h_{1}^{-1}h_{2}=s_{1}\cdots s_{k}$, where
$k=\ell_{H}(h_{1}^{-1}h_{2}),s_{i}\in S.$}$ $\displaystyle=$
$\displaystyle\ell_{G}(c(s_{1},s_{2}\cdots s_{k}h_{2}^{-1}*x)\cdots
c(s_{k},h_{2}^{-1}*x))$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}\ell_{G}(c(s_{i},s_{i+1}\cdots
s_{k}h_{2}^{-1}*x))$ $\displaystyle\leq$
$\displaystyle\kappa\ell_{H}(h_{1}^{-1}h_{2})=\kappa d_{H}(h_{1},h_{2}).$

∎

######  Remark 2.3.

Similar calculation was already used in [kra, lixin]. In fact, there is a
connection between translation-like actions and continuous orbit equivalence
theory, which may be worth exploring further.

##  3\. Applications

As mentioned in the introduction, we use the obstruction to existence of
regular maps, i.e. asymptotic dimension, which was first defined by Gromov
[gromov] (see [a-dimension] for a nice survey), Dirichlet harmonic functions
[harmonic], growth and separation [separation], to deduce results on non-
existence of translation-like actions. Conversely, we may deduce results on
the existence of regular maps between two groups. The above process would
generate many (counter)examples, we just list a few of them here and we refer
the readers to the above papers for definitions of the above invariants.

######  Corollary 3.1.

If $G$ is a finitely generated non-amenable group, then
$F_{2}\rightarrow_{reg}G$, where $F_{2}$ is the non-abelian free group on two
generators.

###### Proof.

This is clear by Whyte’s solution to the geometric von Neumann conjecture, see
[whyte]. ∎

######  Corollary 3.2.

If $G$ is a finitely generated infinite group, then
$\mathbb{Z}\rightarrow_{reg}G$.

###### Proof.

This is clear by Seward’s solution to the geometric Burnside’s problem, see
[seward]. ∎

######  Corollary 3.3.

$\mathbb{Z}^{d}$ does not admit translate-like actions on $F_{2}$ for all
$d\geq 2$.

###### Proof.

By theorem 1.1, we just need to show
$\mathbb{Z}^{d}\not\rightarrow_{reg}F_{2}$ if $d\geq 2$. By [separation,
Corollary 3.3], $\lim_{n\to\infty}sep_{\mathbb{Z}^{d}}(n)=\infty$ if $d\geq
2$, while $sep_{F_{2}}(\cdot)$ is bounded by [separation, Section 2]. Since
the separation function is monotone non-decreasing with respect to regular
maps by [separation, Lemma 1.3], we deduce
$\mathbb{Z}^{d}\not\rightarrow_{reg}F_{2}$. ∎

Corollary 3.3 answers a question of Cohen, see [cohen, p.4]. This question is
one initial motivation for us to look at translation-like actions.

######  Corollary 3.4.

The Baumslag-Solitar group $BS(m,n):=\langle s,t|~{}st^{m}s^{-1}=t^{n}\rangle$
does not admit translate-like actions on the lamplighter group
$(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ for any integers $m,n$.

###### Proof.

Case 1: Assume $mn\neq 0$.

It suffices to show
$BS(m,n)\not\rightarrow_{reg}(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ if $mn\neq
0$.

First, $asdim((\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}))=1$ by [gentimis,
Proposition on p.5]. Then note that the asymptotic dimension is monotone non-
decreasing under regular maps, see [separation, Section 6] or just apply
[a-dimension, Theorem 29] in our setting. Therefore, we are left to show
$asdim(BS(m,n))\geq 2$ if $mn\neq 0$.

First, $BS(m,n)$ is always an infinite group, hence $asdim(BS(m,n))>0$ by
[gentimis, Lemma 1] or Corollary 3.2. We are left to show $asdim(BS(m,n))\neq
1$. By [fuji_whyte, Corollary 1.2] (see also [cohomo.approach, gentimis,
fillinginvariant]), we just need to check $BS(m,n)$ does not contain free
group as a subgroup of finite index. This is clear since $BS(m,n)$ is torsion-
free when $mn\neq 0$ by [kms] and any torsion-free virtually free groups are
free groups by Stallings’ work [stallings].

Case 2: Assume $mn=0$.

$BS(m,n)\cong\mathbb{Z}*(\mathbb{Z}/n\mathbb{Z})$ or
$\mathbb{Z}*(\mathbb{Z}/m\mathbb{Z})$. This group contains a free group, then
it does not admit a translation-like action on
$(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ by [jeandel, Lemma 1.3] and [whyte,
Theorem 6.2]. ∎

######  Remark 3.5.

Note that to prove the above corollary, we can directly focus on $BS(1,n)$,
which is amenable. Also note that if $m=n=1$, $BS(1,1)=\mathbb{Z}^{2}$, then
we can also deduce
$\mathbb{Z}^{2}\not\rightarrow_{reg}(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ by
[separation, Proposition 6.1]. If $m=0$ or $n=0$, then
$BS(m,n)\cong\mathbb{Z}*(\mathbb{Z}/n\mathbb{Z})$ or
$\mathbb{Z}*(\mathbb{Z}/m\mathbb{Z})$, this group has asymptotic dimension one
by [a-dimension, Theorem 84].

Corollary 3.4 answers [jeandel, Conjecture 3] negatively. It also answers the
geometric Gersten problem stated in [seward] in the negative. The original
geometric Gersten problem (stated under a further assumption compared with the
one stated in [seward]) was asked by Whyte in [whyte, p.107]. See [cohen] for
more discussion on this conjecture. Note that the lamplighter group is a
potential counterexample to the above conjectures was already suggested in
[jeandel].

We end the paper with the following question:

######  Question 3.6.

Does [hs, Theorem 2.3] still hold for regular maps (maybe up to some
modification of the definition of fat bigons there)?

Acknowledgments. The author would like to thank his advisor Prof. Hanfeng Li
for constant support and illuminating discussion on this paper. He is also
grateful to Nhan-Phu Chung and Xin Li for sharing their ideas and unpublished
notes related to continuous orbit equivalence theory. He also thanks David
Bruce Cohen for very helpful correspondence.

## References