Domain Independent SVM for Transfer Learning in Brain Decoding
Shuo Zhou Wenwen Li Christopher R. Cox Haiping Lu


SUSY discovery prospects with MoEDAL
    K Sakurai^1, D Felea^2, J Mamuzic^3, N E Mavromatos^4, V A Mitsou^3,∗, J L Pinfold^5, R Ruiz de Austri^3, A Santra^3 and O Vives^3
    Received ; accepted 
======================================================================================================================================




Brain imaging data are important in brain sciences yet expensive to obtain, with big volume (i.e., large p) but small sample size (i.e., small n). To tackle this problem, transfer learning is a promising direction that leverages source data to improve performance on related, target data. Most transfer learning methods focus on minimizing data distribution mismatch. However, a big challenge in brain imaging is  the large domain discrepancies in cognitive experiment designs and subject-specific structures and functions. A recent transfer learning approach minimizes domain dependence to learn common features across domains, via the Hilbert-Schmidt Independence Criterion (HSIC). Inspired by this method, we propose a new Domain Independent Support Vector Machine (DI-SVM) for transfer learning in brain condition decoding. Specifically, DI-SVM simultaneously minimizes the SVM empirical risk and the dependence on domain information via a simplified HSIC. We use public data to construct 13 transfer learning tasks in brain decoding, including three interesting multi-source transfer tasks. Experiments show that DI-SVM's superior performance over eight competing methods on these tasks, particularly an improvement of more than 24% on multi-source transfer tasks.





§ INTRODUCTION

High resolution imaging is playing a major role in progressing science in many basic and applied areas.
In cognitive neuroscience, neuroimaging is used to relate different cognitive functions to patterns of neural activity supported by complex networks of interacting brain regions.
This often takes the form of a classification problem <cit.>.
Machine learning techniques have been applied to sets of neuroimages to distinguish between brain conditions associated with experimental manipulations.
However, while functional magnetic resonance imaging (fMRI; <cit.>) produces volumes with hundreds of thousands of potentially relevant voxels, a typical experiment will have on the order of 100 discrete trials.
This severely constrains the number of training examples available for the classifier.


Transfer learning is a machine learning scheme that can improve the classification performance on a learning task by leveraging the knowledge from related tasks <cit.>.
The task of interest is called the target domain, while the task(s) to be leveraged is called the source domain <cit.>.
A transfer learning problem is homogeneous when the feature and label space of the source and target domains are the same, and heterogeneous if they are different <cit.>.

For the brain decoding problem mentioned above, there are public brain imaging data from multiple sites available, e.g., the OpenNeuro <cit.> and human connectome project (HCP) <cit.>.
For these data, there are many overlapped/similar brain conditions across different cognitive experiments, making homogeneous transfer a promising solution. 
However, existing transfer learning studies on brain decoding <cit.> did not consider aggregating data from multiple experiments with the same/similar labels when training a decoding model.
Here, we make the first attempt, to the best of our knowledge, to investigate homogeneous transfer learning for brain condition decoding.
Homogeneous transfer learning methods are mainly studied in computer vision (CV) and natural language processing (NLP). They focus on minimising data distribution mismatch, i.e., making features from the source and target to have as similar distributions as possible. Two approaches are popular: 1) learning a feature mapping to minimize the distribution mismatch <cit.>; and 2) jointly optimizing the mapping and classifier parameters <cit.>.
However, decoding brain conditions presents challenges different from those in CV/NLP. The same experiment paradigm can have different designs across sites.
Differences in individual brain structures and functions make informative voxels to vary across subjects <cit.>.
Therefore, some methods in neuroscience consider each subject as a learning task to extract subject-specific features <cit.>.
Transfer learning for brain condition decoding needs to take differences in domain information, such as experiment designs and subjects, into account. 

Maximum independence domain adaptation (MIDA) <cit.> introduces a new domain dependence minimization approach to transfer learning. It constructs auxiliary features to encode domain information and learns common cross-domain features with minimum dependence on the domain information, as measured by the Hilbert-Schmidt Independence Criterion (HSIC) <cit.>.
This inspired us to follow a similar approach for brain condition decoding problems, where we can encode different experiment designs and subjects as auxiliary domain information.


In this paper, we propose Domain Independent Support Vector Machine (DI-SVM), a homogeneous transfer classifier for brain condition decoding. It is build on MIDA, which shows that a classifier depending less on domain information can better leverage source data to improve target data prediction. Therefore, we aim to learn a generalized decoding model with minimum dependency on domain information. Specifically, DI-SVM simultaneously minimizes the Support Vector Machine (SVM) empirical risk and the dependence on experiment designs and subjects via a simplified HSIC. This is our first contribution.

Our second contribution is to construct a set of transfer learning problems for brain condition decoding studies. We identify subsets with the same or similar brain conditions from public brain imaging repositories. With the help of a neuroscientist, we carefully design six transfer learning cases (with 13 transfer learning tasks) of increasing difficulties from a psychological perspective. Experimental results show that DI-SVM  outperforms six state-of-the-art transfer learning methods on these tasks, in particular more than 24% improvement on multi-source transfer tasks. This confirms that homogeneous transfer learning is a promising direction for brain condition decoding, and brain sciences more broadly.




§ PRELIMINARIES


Before introducing DI-SVM, we review the HSIC for measuring statistical dependence and three approaches in state-of-the-art transfer learning methods.

Hilbert-Schmidt Independence Criterion (HSIC) is a non-parametric criterion for measuring the statistical dependence between two sets 𝐗={𝐱_i} and 𝐘={𝐲_i}, both with n samples.
HSIC is zero if and only if 𝐗 and 𝐘 are independent. A larger HSIC value suggests stronger dependence.
The empirical HSIC between 𝐗 and 𝐘, ρ_h(𝐗, 𝐘), is given by <cit.>

    ρ_h(𝐗, 𝐘) = 1/(n-1)^2tr(𝐊𝐇𝐋𝐇),

where 𝐊, 𝐇, 𝐋∈ℝ^n× n, 𝐊_i,j := k_x(𝐱_i, 𝐱_j), 𝐋_i,j := k_y(𝐲_i, 𝐲_j), k_x(·), k_y(·) are two kernel functions, e.g., linear, polynomial, or radial basis function (RBF), 𝐇 = 𝐈 -1/n11^⊤ is the centering matrix, and tr(·) is the trace function.
 
Distribution Mismatch Minimization Mapping.
The most popular approach to homogeneous transfer learning is to minimize the distribution mismatch between a source and a target, typically measured by the maximum mean discrepancy (MMD) criterion <cit.>, via learning a mapping of the input data to a subspace. <cit.> proposed Transfer Component Analysis (TCA) to learn such a mapping by minimizing the MMD of marginal distribution mismatch and maximizing the captured variances. <cit.> also proposed Semi-Supervised TCA (SSTCA) for extracting more discriminate features by maximizing HSIC on the data labels.
<cit.> extended TCA to Joint Distribution Adaptation (JDA) to minimize both marginal and conditional distribution mismatch. We can summarize these methods in a general formula

    min_ϕ𝒟(ϕ(𝐗_s), ϕ(𝐗_t)) - Var(ϕ(𝐗)),

where ϕ denotes a feature mapping, 𝐗_s and 𝐗_t are source and target domain data, respectively, 𝒟(ϕ(𝐗_s), ϕ(𝐗_t)) is the distribution mismatch (MMD) between source and target domains, 𝐗 is the combined source and target data, Var(·) denotes the total captured variance. A regularization term on the feature mapping matrix is typically incorporated.
SSTCA has a label dependence objective -μρ_h(ϕ(𝐗), 𝐘) added to Eq. (<ref>), where μ is a hyper-parameter, and 𝐘 is a label matrix (e.g. y_i,j = 1 if 𝐱_i belongs to the jth class; y_i,j = 0 otherwise).



Domain-Invariant Classifier Learning. Another approach learns a classifier by minimizing the prediction error and distribution mismatch jointly. We summarize this approach using the following formula

    min_f, ϕℒ(f(ϕ(𝐗^l), 𝐲)) + λ𝒟(ϕ(𝐗_s), ϕ(𝐗_t)),

where f(·) is a decision function of a classifier, ℒ(f(ϕ(𝐗^l), 𝐲)) is the prediction error, 𝐗^l denotes labeled data, which can be source data only or source data plus target data according to different settings, and 𝐲 is a data label vector. Table <ref> summarizes four such recent methods. 

Domain Dependence Minimization Mapping.
<cit.> proposed the MIDA method using a new approach. It obtains cross-domain features by learning a mapping to minimize the dependence on auxiliary domain information. In contrast, such domain information is not directly modeled in the distribution mismatch minimization mapping or domain-invariant classifier learning approaches.
We summarize the objective of MIDA as

    min_ϕρ_h(ϕ(𝐗), 𝐀) - Var(ϕ(𝐗)),

where 𝐀 encodes the auxiliary domain information. Similar to TCA, MIDA also has a semi-supervised version (SMIDA) that maximizes label dependence by adding a term -μρ_h(ϕ(𝐗), 𝐘) to Eq. (<ref>).



§ DOMAIN INDEPENDENT SVM FOR BRAIN DECODING


In this section, we develop DI-SVM, a new transfer learning method for the brain condition decoding problem. We will first describe the transfer learning setting for this problem and then propose the DI-SVM model and algorithm, followed by some discussions.



 §.§ Transfer Learning Setting for Brain Decoding


We aim to learn a generalized brain condition decoding model from source and target data samples while utilizing the domain information. We consider a semi-supervised setting where all source data samples are labeled while the target data have both labeled samples and unlabeled samples. The objective is to predict the labels of unlabeled target samples.

The target. The target cognitive experiment has n_t fMRI data samples 𝐗_t ∈ℝ^d × n_t of m brain conditions, where d is the dimension of an fMRI data sample, i.e., the number of raw fMRI features. There are ñ_t labeled samples and (n_t - ñ_t) unlabeled samples in the target domain. 

The source. The source consists of data from one or more cognitive experiments with n_s labeled fMRI data samples 𝐗_s ∈ℝ^d × n_s in total, with the same m brain conditions as the target data. 

Domain information encoding. Denote the target and source data jointly as 𝐗 = [𝐗_s,𝐗_t] ∈ℝ^d × n, n = n_s + n_t. Each fMRI sample 𝐱_i (i=1,⋯,n) is collected with a particular experiment design j from a particular subject k, where j=1,⋯,p and k=1,⋯,q, i.e., there are p unique experiment designs and q unique subjects. This is the domain information to be utilized in our transfer learning method, following MIDA <cit.> as reviewed above. We use a simple one-hot-encoding strategy to encode such domain information. Specifically, we construct an experiment design one-hot matrix 𝐄∈ℝ^n× p, where its (i,j)th element e_i,j = 1  if 𝐱_i is collected from experiment j and e_i,j = 0 otherwise.
Similarly, we construct a subject one-hot matrix 𝐒∈ℝ^n× q, where s_i,k = 1  if 𝐱_i is from subject k and s_i,k = 0  otherwise. We then obtain the auxiliary domain information matrix 𝐀∈ℝ^d̂× n by concatenating 𝐄^⊤ and 𝐒^⊤, where d̂ = p+q. 



 §.§ Domain Independent SVM (DI-SVM)


Domain-independent classifier learning. Domain-invariant classifier learning methods (Eq. (<ref>)) minimize prediction errors directly to optimize prediction performance, while domain dependence minimization mapping allows us to leverage important domain information such as the experiment designs and subjects for transfer learning. Here we propose the DI-SVM model by combining the virtues from both approaches, i.e., minimizing prediction error and domain dependence simultaneously. Specifically, we can replace the distribution mismatch term in Eq. (<ref>) with a domain dependence term using HSIC in Eq. (<ref>). This gives the following domain-independent classifier learning model as 

    min_fℒ(f(ϕ(𝐗^l), 𝐲)) + λρ_h(ϕ(𝐗), 𝐀),

where λ>0, 𝐗^l ∈ℝ^d ×ñ denotes all labeled samples from source and target domains, ñ = ñ_t + n_s, and 𝐲 is the label vector. In this paper, we consider only binary labels, i.e., y_i ∈{-1, 1}, i=1,⋯ñ. Accordingly, we consider only m=2, i.e., the classification of two brain conditions.

SVM classifier.
We can plug in any loss function for the first term in our model Eq. (<ref>), such as the square loss, logistic loss, or hinge loss. 
Here we choose the hinge loss for empirical risk minimization in SVM. The linear SVM has a target decision function in a simple form f(𝐱) = β^⊤𝐱, where β is a coefficient vector. The standard linear SVM learns the decision function by solving the following minimization problem

    min_β, ξ1/2β^2+C∑_i^ñξ_i,
    s.t.    y_i β^⊤𝐱_i≤1-ξ_i, y_i ∈{-1, 1},

where ξ_i is the “slack variable” for the ith sample, C is a hyper-parameter controlling the trade-off between more accurate model on training data and larger margin of decision hyperplane, and ñ is the number of labeled sample. The linear SVM can be extended to a nonlinear version by introducing a kernel function k(·). Using the Representer Theorem <cit.>, the ith sample can be represented as 𝐤_i = k(𝐱_i,𝐗), yielding the Kernel SVM

    min_β, ξ1/2β^⊤𝐊β+C∑_i^ñξ_i,
    s.t.    y_i β^⊤𝐤_i≤1-ξ_i, y_i ∈{-1, 1},

where 𝐊 = k(𝐗,𝐗).

Simplified HSIC. 
With the above SVM-based formulation, we can construct a simplified HSIC by viewing the coefficient vector β as a classifier-based feature mapping. This mapping projects input features to a one-dimensional space (i.e., a line), where the projected values represent the label likelihood. Following the principle of dependence minimization, we aim to learn a domain-independent classifier by minimizing the dependence of the label likelihood (projected values) on domain information, i.e., experiment designs and subjects. Therefore, we simplify the HSIC ρ_h(ϕ(𝐗), 𝐀) to the following version

    ρ_sh(ϕ(𝐗), 𝐀)    = tr((β^⊤𝐊)^⊤(β^⊤𝐊)𝐇𝐊_a𝐇)
       = β^⊤𝐊 𝐇𝐊_a𝐇𝐊β,


where 𝐊_a = k_a(𝐀, 𝐀)∈ℝ^n × n, and k_a(·) is a kernel function. Due to the simplicity of 𝐀 (with one-hot encoding), we always use a simple linear kernel here. Our simplified HSIC is constructed directly from classifier coefficient so there is no separate feature mapping step as in STM <cit.> and DMM <cit.>. 

Label recoding.
In a semi-supervised setting of a binary classification problem, a sample 𝐱_i can have a positive label `1', a negative label `-1', or no label, which is denoted as label `0'. Therefore, we recode the label vector 𝐲 into another label vector 𝐲̃, where ỹ_i ∈{-1, 1} if the ith sample is labeled, ỹ_i =0 otherwise, i=1,⋯,n.

DI-SVM model and algorithm.
Using Eqs. (<ref>) and (<ref>), we can formulate the objective function of DI-SVM  as

    min_β, ξ1/2β^⊤𝐊β + C∑_i^nξ_i +  λ/2β^⊤𝐊 𝐇𝐊_a𝐇𝐊β,
    s.t.  ỹ_i β^⊤𝐤_i≤1-ξ_i, ỹ_i ∈{-1, 1, 0}.

This is a quadratic programming (QP) problem that can be solved by standard QP tools.  Algorithm <ref> is the pseudocode of DI-SVM.

The role of unlabeled samples. DI-SVM requires all samples, both labeled and unlabeled, in training. Unlabeled samples can only influence β in the simplified HSIC term, i.e. dependence on domain information. This is because when the ith sample is unlabeled, i.e., ỹ_i=0, we have ỹ_i β^⊤𝐤_i = 0, which means that this sample has no influence on β in the first term of SVM empirical risk.

Relationship with SMIDA. SMIDA is the closest existing model to the proposed DI-SVM. We can view DI-SVM as 1) replacing the label dependence term ρ_h(ϕ(𝐗), 𝐘) in SMIDA with an SVM empirical risk term (hinge loss function), and 2) learning a mapping to a one-dimensional classification space (i.e., a line) rather than a low-dimensional subspace.






§ EXPERIMENTS

To evaluate DI-SVM, we design six transfer learning cases with 13 transfer learning tasks for brain decoding, built from six datasets, and compare DI-SVM against eight competing methods. 







	





Dataset selection. With the help of a neuroscientist, we carefully selected six datasets (A to F) that are most meaningful from psychological perspective from the public OpenfMRI repository.[<https://legacy.openfmri.org> or <https://openneuro.org>.] Each dataset is from an experiment, which is treated as a domain. Table <ref> summarizes basic information on these datasets. Each dataset contains fMRI time-series data for multiple subjects. Subjects from the same accession number (ds×××) are the same and there is no overlapped subject between accession numbers. There are two brain conditions selected from each dataset. Each is considered as a class and has the same number of samples. Thus, we have binary classification problems that discriminate between brain conditions in an experiment.


Preprocessing. Each sample was preprocessed with the protocol published by <cit.> to obtain the Z-score statistical parametric map (SPM) of size 91 × 109 × 91, which is then reduced to a vector of size 228,546 containing only the valid voxels (some are outside of the brain).





Six transfer learning cases. With the help of a neuroscientist, we constructed six distinct cases of transfer learning problems with increasing difficulties, according to the paradigms used (whole experimental processes), subjects involved, cognitive control demands, and response types. Table <ref> summarizes these six cases. 

Case 1: Same paradigm, subject, and control, different responses.
The first case aims to transfer between two stop signal tasks completed by the same subjects.
These tasks involve producing conditioned responses to simple stimuli, but inhibiting the response on 25% of trials preceded by a stop signal.
The latency between the stop signal and stimulus onset determines the difficulty in inhibiting the response, and latency was adjusted during the experiment so that subjects failed to inhibit about 50% of the time.
Both tasks required verbal responses and involved the same cognitive control demands, but differed in the linguistic complexity of the response: 1) the stimulus were strings of letters that were not English words but could be “sounded out” and the response was to pronounce each string as an English pseudo-word; 2) the stimulus were single letters and the response was to name them aloud. Successful and unsuccessful stops are the two brain conditions to classify.


Cases 2 & 3: Same paradigm and subjects, different controls.
Both cases aim to transfer between each of the stop signal tasks from Case 1 and a nearly identical task with the same subjects, but requiring a manual response (button press with index or middle finger) rather than a verbal response. Again, the two brain conditions to classify are successful and unsuccessful stops.

Case 4: Same paradigm, different subjects and controls.
This case studies the transfer between experiments A and D, which were conducted on different subjects. The primary cognitive difference is that the conditional stop signal in experiment D should only be heeded for one of the two stimuli.
Thus, these tasks differed in their cognitive control demands and response complexity, yet share the same general “stop signal” paradigm.
This case classified successful against unsuccessful stops as well.


Case 5: Different paradigms and subjects, similar controls. 
This case studies the transfer between the flanker task (E) and the Simon task (F).
They are distinct paradigms that differ in how stimulus are presented, the kinds of stimuli involved, and the source of task interference to be inhibited.
Yet, they share abstract similarity: they both involve overcoming a prevalent response bias and are understood to involve similar executive cognitive functions. The two brain conditions to classify correspond to congruent and incongruent trials.

Case 6: Multi-source transfer with the same subjects.
Cases 1–5 study transfer learning on increasingly difficult problems, where one source domain is leveraged to improve classification in a target domain.
This last case studies whether leveraging multiple source domains can obtain further improvement. From the three stop signal tasks introduced in Cases 1–3, we treat each possible pair as the source domain for the remaining task (as the target domain). 


In total, there are 13 transfer learning tasks constructed.

Eight methods compared. We evaluate DI-SVM against eight methods: two simple baselines 1) PCA_t, where PCA was only performed on the target data, 2) PCA_s+t, where PCA was performed on the combined source and target data; and six state-of-the-art transfer learning methods discussed in Section <ref>: 3) TCA, 4) SSTCA, 5) JDA, 6) MEDA, 7) MIDA, and 8) SMIDA.
PCA, TCA, SSTCA, JDA, MIDA, and SMIDA only learn a feature mapping and they use SVM as the classifier. Both linear and RBF kernels were studied for such SVM classifiers, and also MEDA and DI-SVM. 
We will report the accuracy obtained by DI-SVM with both linear and RBF kernels. 
For other methods, we report the accuracy from the best performing kernel. 

Experimental Settings.
We performed 10 × 5-fold cross-validation on the target domain. For each split, the target training samples and all source samples (except for PCA_t) were used for training, and the remaining target test samples are for testing. On each training set, we determined the best hyper-parameters via grid search with 20 further random splits (20% for validation and 80% for training).
For DI-SVM, we first fixed λ = 1 and searched for the best C in [10^-3, 10^4] on regular grids of log scale with a step size of one (i.e., eight values). Then we fixed the best C and searched for the best λ among five values only: {0.01, 0.1, 1, 10, 100}. Sensitivity studies will be provided in the end of this section.

We also tuned hyper-parameters for all the other methods.
For PCA, TCA, SSTCA, JDA, MIDA and SMIDA, we searched for the best subspace dimension from {20, 40, 50, 60 ,80, 100}.  
For TCA, SSTCA, JDA, MIDA and SMIDA, linear kernel was used. 
For SVM, we searched for the best C ∈ [10^-3, 10^4], and the best Gamma value in [10^-6, 10^2] when using the RBF kernel.
For other method-specific hyper-parameters, we followed the strategies stated in their original papers.


We used one-hot encoding as presented in Section <ref> to obtain the domain information matrix 𝐀 for DI-SVM, MIDA, and SMIDA. DI-SVM, SSTCA, and SMIDA also intake the label vector 𝐲̃ (see Section <ref>). Linear kernel was used to construct the label kernel matrix for SSTCA and SMIDA, i.e. 𝐊_y = ỹ^⊤ỹ.


Experimental Results and Discussions.




Tables <ref> and <ref> summarize the decoding accuracy of ten single-source transfer learning tasks and three multi-source transfer learning tasks respectively, with both the mean and standard deviation reported. The best results are highlighted in bold and the second best ones are underlined. We have five observations:

	
  * DI-SVM outperformed all comparing methods, though the improvement decreased from easy to difficult cases. This indicates that more difficult cases from psychological perspective are also more difficult for transfer learning.
	
  * For DI-SVM, multi-source transfer did not always outperform single-source transfer. The accuracy of transfer learning tasks, A&C→B and B&C→A are both higher than the corresponding single-source tasks A→B and C→B, and B→A and C→A, respectively. However, the accuracy of A&B→C is lower than that of  B→C. This indicates that source selection can have large influence the performance. Nevertheless, DI-SVM outperformed all the other methods significantly (by more than 24% with the linear kernel) on the multi-source transfer tasks. Therefore, when there is no clear preference of a particular source, multi-source transfer is a preferred choice.
	
  * HSIC-based methods MIDA and DI-SVM have outperformed other transfer learning methods in both single-source and multi-source transfer tasks. This shows the effectiveness of using domain information (via one-hot encoding) and performing dependence minimization on it (via HSIC). 
	
  * PCA_s+t outperformed PCA_t in Cases 1 to 3, with all subjects common, but PCA_t outperformed PCA_s+t in Cases 4 and 5, with no subject in common.
	This demonstrates the large influence of domain information (subjects) on the classification accuracy. 
	
  * In single-source transfer tasks, TCA and MIDA outperformed SSTCA and SMIDA, respectively, which may seem counter-intuitive because semi-supervised learning has more information (the labels) utilized. In contrast, in multi-source transfer tasks where there were more training samples, SSTCA slightly outperformed TCA on the whole, and the performance difference between SMIDA and MIDA was much smaller than in single-source transfer tasks. A possible explanation is that in small sample setting, maximizing dependence on training data labels is more susceptible to overfitting resulting poorer generalization performance. 


Hyper-parameter Sensitivity.

We evaluated the sensitivity of DI-SVM with linear kernel against hyper-parameters C and λ under five-fold cross validation.
Figure <ref> shows the sensitivity against C ∈ [10^-3, 10^4] when fixing λ = 1. We can observe that the accuracy stays stable when C ≤ 1, and shows a trend of decreasing when C ∈ [10^0, 10^4]. Since a smaller value of C can lead to a larger SVM classification margin, we expect a classifier with a larger margin to generalize better and have higher prediction accuracy.
Figure <ref> shows the sensitivity against λ∈{0, 0.01, 0.1, 1, 10, 100} when fixing C = 1. We can observe that the prediction accuracy of DI-SVM kept almost constant when λ> 0 and it was not sensitive to λ. When λ = 0, i.e., without minimizing domain dependence, DI-SVM becomes a standard Kernel SVM and the performance was poorer.



§ CONCLUSION

In this paper, we proposed Domain Independent SVM (DI-SVM) for transfer learning in brain decoding. It works by minimizing the SVM empirical risk and dependence on auxiliary domain information measured by a simplified HSIC. We evaluated DI-SVM against PCA and six state-of-the-art transfer learning methods on 13 transfer learning tasks. Experimental results showed the superior overall performance of DI-SVM over other methods, particularly on multi-source transfer with a 24% improvement. This confirmed the benefits of leveraging auxiliary domain information and HSIC in transfer learning for brain condition classification.



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general thesis and conclusion of your paper. A reader should be able
to learn the purpose of the paper and the reason for its importance
from the abstract. The abstract should be no more than 200 words long.



 §.§ Text


The main body of the text immediately follows the abstract. Use
10-point type in a clear, readable font with 1-point leading (10 on
11).

Indent when starting a new paragraph, except after major headings.



 §.§ Headings and Sections


When necessary, headings should be used to separate major sections of
your paper. (These instructions use many headings to demonstrate their
appearance; your paper should have fewer headings.). All headings should be capitalized using Title Case.



  §.§.§ Section Headings


Print section headings in 12-point bold type in the style shown in
these instructions. Leave a blank space of approximately 10 points
above and 4 points below section headings.  Number sections with
arabic numerals.



  §.§.§ Subsection Headings


Print subsection headings in 11-point bold type. Leave a blank space
of approximately 8 points above and 3 points below subsection
headings. Number subsections with the section number and the
subsection number (in arabic numerals) separated by a
period.



  §.§.§ Subsubsection Headings


Print subsubsection headings in 10-point bold type. Leave a blank
space of approximately 6 points above subsubsection headings. Do not
number subsubsections.



  
Titled paragraphs. You can use titled paragraphs if and 
only if the title covers exactly one paragraph. Such paragraphs must be
separated from the preceding content by at least 3pt, and no more than
6pt. The title must be in 10pt bold font and ended with a period. 
After that, a 1em horizontal space must follow the title before 
the paragraph's text.

In  titled paragraphs must be typeset using

\paragraph{Title.} text .




  §.§.§ Acknowledgements


You may include an unnumbered acknowledgments section, including
acknowledgments of help from colleagues, financial support, and
permission to publish. If present, acknowledgements must be in a dedicated,
unnumbered section appearing after all regular sections but before any
appendices or references.

Use 

    \section*{Acknowledgements})

to typeset the acknowledgements section in .



  §.§.§ Appendices


Any appendices directly follow the text and look like sections, except
that they are numbered with capital letters instead of arabic
numerals. See this document for an example.



  §.§.§ References


The references section is headed “References”, printed in the same
style as a section heading but without a number. A sample list of
references is given at the end of these instructions. Use a consistent
format for references. The reference list should not include unpublished
work.



 §.§ Citations


Citations within the text should include the author's last name and
the year of publication, for example <cit.>.  Append
lowercase letters to the year in cases of ambiguity.  Treat multiple
authors as in the following examples: <cit.>
or <cit.> (for more than two authors) and
<cit.> (for two authors).  If the author
portion of a citation is obvious, omit it, e.g.,
Nebel nebel:jair-2000.  Collapse multiple citations as
follows: <cit.>.
abelson-et-al:scheme
bgf:Lixto
brachman-schmolze:kl-one
gottlob:nonmon
gls:hypertrees
levesque:functional-foundations
levesque:belief
nebel:jair-2000



 §.§ Footnotes


Place footnotes at the bottom of the page in a 9-point font.  Refer to
them with superscript numbers.[This is how your footnotes
should appear.] Separate them from the text by a short
line.[Note the line separating these footnotes from the
text.] Avoid footnotes as much as possible; they interrupt the flow of
the text.



§ ILLUSTRATIONS


Place all illustrations (figures, drawings, tables, and photographs)
throughout the paper at the places where they are first discussed,
rather than at the end of the paper.

They should be floated to the top (preferred) or bottom of the page, 
unless they are an integral part 
of your narrative flow. When placed at the bottom or top of
a page, illustrations may run across both columns, but not when they
appear inline.

Illustrations must be rendered electronically or scanned and placed
directly in your document. All illustrations should be understandable when printed in black and
white, albeit you can use colors to enhance them. Line weights should
be 1/2-point or thicker. Avoid screens and superimposing type on
patterns as these effects may not reproduce well.

Number illustrations sequentially. Use references of the following
form: Figure 1, Table 2, etc. Place illustration numbers and captions
under illustrations. Leave a margin of 1/4-inch around the area
covered by the illustration and caption.  Use 9-point type for
captions, labels, and other text in illustrations. Captions should always appear below the illustration.



§ TABLES


Tables are considered illustrations containing data. Therefore, they should also appear floated to the top (preferably) or bottom of the page, and with the captions below them.





If you are using , you should use the booktabs package, because it produces better tables than the standard ones. Compare Tables <ref> and <ref>. The latter is clearly more readable for three reasons:


    
  * The styling is better thanks to using the booktabs rulers instead of the default ones.
    
  * Numeric columns are right-aligned, making it easier to compare the numbers. Make sure to also right-align the corresponding headers, and to use the same precision for all numbers.
    
  * We avoid unnecessary repetition, both between lines (no need to repeat the scenario name in this case) as well as in the content (units can be shown in the column header).




§ FORMULAS


IJCAI's two-column format makes it difficult to typeset long formulas. A usual temptation is to reduce the size of the formula by using the small or tiny sizes. This doesn't work correctly with the current  versions, breaking the line spacing of the preceding paragraphs and title, as well as the equation number sizes. The following equation demonstrates the effects (notice that this entire paragraph looks badly formatted):



    x = ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i



Reducing formula sizes this way is strictly forbidden. We strongly recommend authors to split formulas in multiple lines when they don't fit in a single line. This is the easiest approach to typeset those formulas and provides the most readable output


    x =   ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + 
    
        +    ∏_i=1^n ∑_j=1^n j_i


If a line is just slightly longer than the column width, you may use the resizebox environment on that equation. The result looks better and doesn't interfere with the paragraph's line spacing: 

    .91!x = ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i


This last solution may have to be adapted if you use different equation environments, but it can generally be made to work. Please notice that in any case:


    
  * Equation numbers must be in the same font and size than the main text (10pt).
    
  * Your formula's main symbols should not be smaller than small text (9pt).


For instance, the formula


    .91!x = ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j + ∏_i=1^n ∑_j=1^n j_i + ∏_i=1^n ∑_j=1^n i_j


would not be acceptable because the text is too small.



§ ALGORITHMS AND LISTINGS


Algorithms and listings are a special kind of figures. Like all illustrations, they should appear floated to the top (preferably) or bottom of the page. However, their caption should appear in the header, left-justified and enclosed between horizontal lines, as shown in Algorithm <ref>. The algorithm body should be terminated with another horizontal line. It is up to the authors to decide whether to show line numbers or not, how to format comments, etc.

In  algorithms may be typeset using the algorithm and algorithmic packages, but you can also use one of the many other packages for the task.  





§ ACKNOWLEDGMENTS


The preparation of these instructions and the  and Bib
files that implement them was supported by Schlumberger Palo Alto
Research, AT&T Bell Laboratories, and Morgan Kaufmann Publishers.
Preparation of the Microsoft Word file was supported by IJCAI.  An
early version of this document was created by Shirley Jowell and Peter
F. Patel-Schneider.  It was subsequently modified by Jennifer
Ballentine and Thomas Dean, Bernhard Nebel, Daniel Pagenstecher,
Kurt Steinkraus, Toby Walsh and Carles Sierra. The current version 
has been prepared by Marc Pujol-Gonzalez and Francisco Cruz-Mencia.





§  AND WORD STYLE FILES


The  and Word style files are available on the IJCAI–19
website, <http://www.ijcai19.org>.
These style files implement the formatting instructions in this
document.

The  files are ijcai19.sty and ijcai19.tex, and
the Bib files are named.bst and ijcai19.bib. The
 style file is for version 2e of , and the Bib
style file is for version 0.99c of Bib (not version
0.98i). The ijcai19.sty style differs from the ijcai18.sty file used for IJCAI–18.

The Microsoft Word style file consists of a single file, ijcai19.doc. This template differs from the one used for
IJCAI–18.

These Microsoft Word and  files contain the source of the
present document and may serve as a formatting sample.  

Further information on using these styles for the preparation of
papers for IJCAI–19 can be obtained by contacting pcchair@ijcai19.org.



named


