Similarity Learning with Higher-Order Proximity for Brain Network Analysis

Similarity Learning with Higher-Order Proximity for Brain Network Analysis

Guixiang Ma1, Nesreen K. Ahmed2, Ted Willke2, Dipanjan Sengupta2, Michael W. Cole3, Nick Turk-Browne4, Philip S. Yu1
1University of Illinois at Chicago, 2Intel Labs, 3Rutgers University, 4Yale University
gma@cs.uic.edu, nesreen.k.ahmed@intel.com, ted.willke@intel.com, dipanjan.sengupta@intel.com
mwcole@mwcole.net, nicholas.turk-browne@yale.edu, psyu@cs.uic.edu
Abstract

In recent years, the similarity learning problem has been widely studied. Most of the existing works focus on images and few of these works could be applied to learn similarity between neuroimages, such as fMRI images and DTI images, which are important data sources for human brain analysis. In this paper, we focus on the similarity learning for fMRI brain network analysis. We propose a general framework called “Multi-hop Siamese GCN” for similarity learning on graphs. This framework provides multiple options for refining the graph representations with high-order structure information, thus can be used for graph similarity learning on various brain network data sets. We apply the proposed Multi-hop Siamese GCN approach on four real fMRI brain network datasets for similarity learning with respect to brain health status and cognitive abilities. Our proposed method achieves an average AUC gain of compared to PCA, and an average AUC gain of compared to S-GCN across a variety of datasets, indicating its promising learning ability for clinical investigation and brain disease diagnosis.

1 Introduction

In many applications, the ability to compute similarity scores between objects is crucial to a variety of machine learning tasks such as classification, clustering and ranking. For example, finding images that are similar to a query image is an indispensable problem in search engines[\citeauthoryearWang et al.2014], and an effective image similarity metric is the key for finding similar images. In the past decade, quite a few of works have been done on similarity learning. In [\citeauthoryearBautista, Sanakoyeu, and Ommer2017], similarity learning is formulated as a partial ordering task with soft correspondences of all samples to classes. Adopting a strategy of self-supervision, a CNN is trained to optimally represent samples in a mutually consistent manner while updating the classes. The similarity learning and grouping procedure are integrated in a single model and optimized jointly. Another deep learning model for ranking is proposed in [\citeauthoryearWang et al.2014] for learning fine-grained image similarity. An efficient triplet sampling algorithm is proposed to learn the model with distributed asynchronous stochastic gradient. The proposed approach is shown to outperform models based on hand-crafted visual features and deep classification models.

Although the similarity learning problem has been widely studied, most existing works focus on images and few of these works could be applied to learn similarity in neuroimaging, such as fMRI images and DTI images, which are important data sources for human brain analysis. In this paper, we focus on the similarity learning for fMRI brain data analysis. Instead of looking at the original fMRI images, we look at the fMRI brain connectivity networks derived from the original fMRI images, as the brain connectivity networks could reflect the overall region-by-region interactions in human brain and have be shown to be an important view to investigate human brains [\citeauthoryearBullmore and Sporns2009]. Existing works in this area have shown that the structure information in human brain networks could reflect their brain activity patterns, and people with brain disorders tend to have different patterns with healthy people. The graph representation has privileges in capturing structure and high-order information of human brain. Therefore, we study the similarity learning problem in the graph domain and aim to apply it for fMRI brain connectivity (network) analysis.

Recently, graph convolutional neural networks (GCNs) have drawn much attention to learn useful representations from graph data, and they have been shown to be effective compared to other relational learning methods [\citeauthoryearDefferrard, Bresson, and Vandergheynst2016, \citeauthoryearLi, Han, and Wu2018]. However, these works mainly focus on social and information networks, where the goal is the node-level similarity or relationship analysis. In this paper, however, we focus on learning useful representations from fMRI brain connectivity networks. Our goal is to model the structural similarity for multi-subject neurological or cognitive status analysis. Since there are various structure information in graphs and the brain networks might differ in global structure or local structure, the main challenge of this similarity learning task is how to build a general learning framework that can learn discriminating graph structure features from graphs using GCNs and leverage the graph structure for the similarity learning task. Recently, [\citeauthoryearKtena et al.2018] studied the similarity learning problem on graphs with GCNs and applied to fMRI brain networks, which is the most relevant work to our work in this paper. However, they did not consider high-order information of graphs when applying GCNs, thus may ignore important graph features in the similarity learning process. In this paper, we propose to incorporate the high-order structural information of graphs into the GCNs, and build a general framework with Siamese network for similarity learning of graphs. Specifically, we apply a random walk strategy with sliding windows to capture high-order information on graphs and use it to refine the graph representations, which allows for multi-hop convolutions on graphs. Our contributions can be summarized as follows:

  • We propose a general framework called ”Multi-hop Siamese GCN” for similarity learning on graphs. This framework provides multiple options for refining the graph representations with high-order structure information, thus can be used for graph similarity learning on various data sets.

  • The proposed framework employs random walks with sliding windows to obtain high-order information on graphs and leverage the high-order information for the similarity learning task.

  • We apply the Multi-hop Siamese GCN approach on four real fMRI brain network datasets for similarity learning with respect to brain health status and cognitive abilities. The experiment results demonstrate the effectiveness of the proposed framework for similarity learning in brain network analysis.

  • Our proposed approach achieves an average AUC gain of % compared to PCA, and an average AUC gain of % compared to S-GCN across a variety of datasets, indicating its promising learning ability for clinical investigation and applications.

Figure 1: The Multi-hop Siamese GCN framework

2 Preliminaries

In many machine learning problems where data comes in graphs, a key application is how to measure the similarity between graphs. In the field of brain network analysis, measuring similarity between brain networks is especially important for further analysis such as brain disorder diagnosis. Existing methods for similarity estimation between graphs are mainly based on graph embedding, graph kernels or motifs [\citeauthoryearLivi and Rizzi2013]. These methods are designed for specific scenarios and have their limitations. For example, the graph embedding learned in [\citeauthoryearAbraham et al.2017] may discard structural information that could be important for similarity estimation. In [\citeauthoryearTakerkart et al.2014], the graph kernels used for brain network comparison focus on features of small subgraphs, which ignored global structures of graph. Another problem in these works is that the graph feature extraction and similarity estimation are done in completely separate stages, where the features extracted may not be suitable for similarity estimation.

Recently, a metric learning method with spectral graph convolutions is proposed in [\citeauthoryearKtena et al.2018], where a Siamese network with graph convolutional neural networks is used to get similarity estimate of two brain connectivity networks. This method consists of two components: (a) Graph Convolutional Network(GCN) and (b) Siamese Network. This method shows promising results in the experiment of similarity estimate between brain connectivity networks. However, the way that GCN was used in that work focuses on local structure in the graph representation, while ignoring high-order structural information, which makes the method less generic. In this paper, we go beyond of that method and propose a Multi-hop Siamese GCN approach, which is a general framework with different options in each component, and the Multi-hop property of the approach allows us to incorporate high-order structural information from the graph representations into the learning process. In this section, we will first introduce GCN and Siamese network individually. Then we will introduce the proposed Multi-hop Siamese GCN approach, which is a general framework with different options in each component for similarity learning of graphs.

2.1 Graph Convolutional Networks

Graph convolutional network (GCN), as a generalized convolutional neural network from grid-structure domain to graph-structure domain, has been emerging as a powerful approach for graph mining [\citeauthoryearBruna et al.2013, \citeauthoryearDefferrard, Bresson, and Vandergheynst2016]. In GCNs, filters are defined in the graph spectral domain. Given a graph , where is the set of vertices, is the set of edges, and is the adjacency matrix, the diagonal degree matrix will have elements . The graph Laplacian matrix is , which can be normalized as , where is the identity matrix. As is a real symmetric positive semidefinite matrix, it has a set of orthonormal eigenvectors , and their associated eigenvalues . The Laplacian is diagonalized by the Fourier basis and where . The graph Fourier transform of a signal can then be defined as [\citeauthoryearShuman et al.2013]. The transform enables the convolution operation on graph in the Fourier domain. Suppose a signal vector is defined on the nodes of graph , where is the value of at the node. Then the signal can be filtered by as

(1)

where the filter can be defined as and the parameter is a vector of Fourier coefficients[\citeauthoryearShuman et al.2013].

In [\citeauthoryearBruna et al.2013], GCN was formulated for the first time, which was parameterised on the eigenvectors of Laplacian. However, their computations of the eigendecompostition are very expensive, and the filters represented in the spectral domain may not be localized in the graph spatial domain. To overcome these issues, [\citeauthoryearDefferrard, Bresson, and Vandergheynst2016] proposed to use a polynomial filter by

(2)

where the parameter is a vector of polynomial coefficients. According to [\citeauthoryearDefferrard, Bresson, and Vandergheynst2016], the above filter is exactly -localized, which means the nodes with shortest path length greater than are not considered for the convolution. To further reduce computational complexity, [\citeauthoryearDefferrard, Bresson, and Vandergheynst2016] proposed to use the Chebyshev polynomials which can be computed recursively by with and , and a filter of order is parameterized as the truncated expansion

(3)

Then filtering operation can be written as , where is the Chebyshev polynomial of order k with the Laplacian . The output feature map of sample s is then given by

(4)

where are the input feature maps, and represents the number of input filters. The vectors of Chebyshev coefficients are the layer’s trainable parameters.

A graph convolutional neural network can therefore be built by stacking multiple convolutional layers in the form of Eq. 4, with a non-linearity following each layer.

2.2 Siamese Network

Siamese networks were first introduced in [\citeauthoryearBromley et al.1994] to solve image matching problem for signature verification. A Siamese network consists of two twin neural networks who share parameters with each other. The inputs of the twin networks are distinct, but their highest-level feature representations are joined by a function at the top. As the parameters between the twin networks are tied, each input will be processed in the same way respectively in the twin networks, which can guarantee that similar input samples not be mapped to very different locations by the respective networks. Therefore, the Siamese network tend to be good for differentiating the two inputs or measuring the similarity between them. Some existing works on image similarity learning use CNNs in the twin networks of Siamese, as CNNs works well in learning 2D grid features from images[\citeauthoryearChopra, Hadsell, and LeCun2005]. In this work, as we aim to learn similarity metric between graphs, we explore the capability of GCNs with Siamese architecture. Figure. 2 shows a simple illustration of the Siamese architecture with GCNs in the twin networks.

2.3 Multi-hop Siamese GCN: A General Framework

As introduced above, GCN uses spectral filterings, which consider localized convolutions while ignoring the nodes with shortest path length beyond a threshold. However, high-order structural information are very important for learning from graphs [\citeauthoryearRossi, Ahmed, and Koh2018, \citeauthoryearBenson, Gleich, and Leskovec2016, \citeauthoryearRossi, Zhou, and Ahmed2017]. In this section, we introduce a general framework for similarity learning of graphs, which is able to capture high-order structure of graphs and incorporate it into the similarity learning process. Figure 1 shows the framework we propose. The detailed illustration of the proposed framework is as follows.

Problem Definition.

Given a pair of graphs and , where and contain the same number of vertices with fully aligned physical meanings, the goal of similarity learning on the pair of graphs is to learn a similarity score between and .

Random Walk Sampling.

Random walk, as a way for sampling, has been used for sampling vertexes or edges for graphs, and the sampling on graphs tend to capture community structure information[\citeauthoryearPerozzi, Al-Rfou, and Skiena2014, \citeauthoryearAhmed et al.2018]. In this paper, we employ a random walk sampling process on graphs and aim to refine the graph representations with the high-order structural information obtained by the random walks, which will be further incorporated into the similarity learning process in the framework. We denote a random walk rooted at vertex as , which is a stochastic process with random variables , and is a vertex chosen randomly from the neighbors of vertex . Given a graph , the random walk generator samples uniformly a random vertex as the root of the random walk. The walk uniformly samples a vertex from the neighbors of the root, after which it continues sampling from the neighbors of the last vertex visited until the maximum path length is reached. There could be multiple walks starting from each vertex, depending on the number of walks specified[\citeauthoryearPerozzi, Al-Rfou, and Skiena2014]. Line in Algorithm 1 illustrates the random walk sampling process we employ for capturing high-order structure information. Note that we slide a window with size on each walk generated and record the frequency of nodes that cooccur within a window in , and decides the number of hops considered for refining the graph representation.

Figure 2: Siamese Architecture with Graph Convolutional Networks

Multi-hop Siamese GCN.

Algorithm 1 shows the overall process of the Multi-hop Siamese GCN approach. As the spectral graph convolutional networks filter signals are defined on a common graph structure for all samples, we first estimate the mean functional connectivity matrix among the training samples by computing the mean adjacency matrix over the graphs and obtain the average graph . In order to refine the graph representation, we apply random walk on , after which we slide a window with size on each walk to get the cooccurrence frequency between two nodes and record the frequency in the matrix . After the random walk sampling stage, we obtain a -nn graph, which is the refined graph encoded with multi-hop high-order structure of the original graph. We compute the Laplacian matrix of the refined graph and denote it with . Now we start the model learning of Multi-hop Siamese GCN. We first prepare the pairs of training samples from with label for each pair of same class and for each pair with different classes. We also initialize the neural network parameters of the GCNs with Siamese network. Then we input the pairs into the Siamese GCNs and perform spectral convolutions, after which the outputs of the twin GCNs are combined and a similarity estimiate will be obtained for each pair. Then we compute the loss for the Siamese network. We use the Hinge loss in Equation (5). To optimize the model, we apply the stochastic gradient descent with Adaptive Moment Estimation (ADAM) algorithm proposed in [\citeauthoryearKingma and Ba2014]. After the training process, we will have a well trained Multi-hop Siamese GCN model for learning similarity scores between a pair of graphs.

(5)
1: (training graph samples); (class labels); random walk parameters: (number of walks), (walk length), (window size)
2:Compute for ;
3:Initialize a frequency matrix with s;
4:for  to  do
5:      Shuffle;
6:     for each  do
7:          ;
8:          Update ;
9:     end for
10:end for
11:Obtain a -nn graph based on ;
12:Add the edges of the -nn graph into ;
13:Obtain the refined adjacency matrix ;
14:Prepare pairs of training samples from ;
15:Initialize the parameters of GCNs in Siamese network;
16:while not converge do
17:     Perform spectral filterings according to Equation (4);
18:     Compute the similarity estimate for the pair;
19:     Compute the loss according to Equation (5) ;
20:     Apply stochastic gradient descent with ADAM optimizer to update ;
21:end while
Algorithm 1 Multi-hop Siamese GCN

3 Experiments & Results

To evaluate the performance of the proposed model for similarity learning of brain networks, we test our framework on four real resting-state fMRI brain datasets and compare with state-of-the-art baselines.

3.1 Datasets and Preprocessing

  • Autism Brain imaging Data Exchange (ABIDE): This dataset is provided by the ABIDE initiative [\citeauthoryearDi Martino et al.2014]. It has the resting-state fMRI images of 70 patients with autism spectrum disorder (ASD) and 102 healthy controls, acquired from the largest data acquisition site involved in that project. The preprocessing of the fMRI data includes slice timing correction, motion correction, band-pass filtering and registering to standard anatomical space. After the preprocessing, a brain network with 264 nodes was constructed for each subject by computing the pearson correlation between the fMRI time series of the 264 putative regions.

  • Human Connectome Project (HCP): This dataset consists of resting-state fMRI imaging data and behavioral data of 100 healthy volunteers from the publicly available Washington Univeristy - Minnesota (WU-Min) Humman Connectome Project (HCP) [\citeauthoryearVan Essen et al.2013]. The preprocessing of the fMRI data consists of intensity normalization, phase-encoding direction unwarping, motion, correction, spatial normalization to standard template and artifact removing[\citeauthoryearSpronk et al.2018]. After preprocessing, for each subject, BOLD time series were extracted from the 360 parcels, and functional connectivity network with 360 nodes was constructed for each individual. In this work, we are interested in solving the pair classification problem based on the label from cognitive traits.

  • Bipolar: This dataset consists of the fMRI data of 52 bipolar I subjects who are in euthymia and 45 healthy controls with matched demographic characteristics. The brain networks were constructed with the CONN111http://www.nitrc.org/projects/conn toolbox [\citeauthoryearWhitfield-Gabrieli and Nieto-Castanon2012]. The raw images were realigned and co-registered, followed by the normalization and smoothing steps. Then the confound effects from motion artifact, white matter, and CSF were regressed out of the signal. Finally, the brain networks were created using the signal correlations between each pair of regions among the 82 labeled Freesurfer-generated cortical/subcortical gray matter regions.

  • Human Immunodeficiency Virus Infection (HIV): This dataset is collected from the Chicago Early HIV Infection Study at Northwestern University[\citeauthoryearRagin et al.2012]. It contains the resting-state fMRI data of 77 subjects, 56 of which are early HIV patients and the other 21 subjects are seronegative controls. We use the DPARSF toolbox222http://rfmri.org/DPARSF. to process the fMRI data. The images were realigned to the first volume, followed by the slice timing correction and normalization. We focus on the 116 anatomical volumes of interest (AVOI), corresponding to 116 brain regions. Then we construct a brain network with the 90 cerebral regions, where each node in the graph represents a cerebral region, and links are created based on the correlations between different regions.

3.2 Baselines and Metrics

We compare our Multi-hop Siamese GCN framework with two other baseline methods for two classification tasks based on the similarity learning on brain networks. We use classification AUC and accuracy as the evaluation metrics.

  • PCA is the Principal Component Analysis approach that is widely used for dimension reduction and feature extraction [\citeauthoryearSmith2002]. We apply PCA on the correlation matrices of the brain networks and perform similarity learning based on the PCA results.

  • Siamese GCN (S-GCN) is the method proposed in [\citeauthoryearKtena et al.2018], which learns similarity scores between graphs based on the outputs of graph convolutional neural networks in a Siamese framework. This was the first work of applying graph convolutional neural network on brain connectivity networks for similarity learning.

  • Multi-hop Siamese GCN (S-MGCN) is the proposed approach in this paper, which is a general framework that provides options for incorporating high-order information of graphs into the similarity learning process.

In the evaluations of the Siamese GCN and Multi-hop Siamese GCN, we use of the data for training and the other for testing. We choose the optimal parameter settings for the Siamese GCN model following the instructions provided in [\citeauthoryearKtena et al.2018]. For the PCA model, as it is an unsupervised method, we directly apply it on the testing portion of the data. The only parameter in the PCA model is the number of components to be preserved in the output lower dimensional representation. We use the number of atlas regions (i.e.,110) as the number for the ABIDE dataset and HCP dataset, as the brain networks in two datasets involves over 200 brain regions and we hope to keep the brain network aligned with each other while doing PCA by latently mapping them to the 110 atlas regions. For the HIV and HCP datasets, we use for the number of components, which is equal to the number of the ICA non-artefactual components introduced in [\citeauthoryearKtena et al.2018]. After we obtain the output representations from PCA, we calculate the similarity score for each pair according to Equation (6) [\citeauthoryearFrey and Dueck2007].

(6)

where and are the PCA results of subject and , respectively. For each experiment, we run for times and report the average results.

3.3 Evaluation

We evaluate the performance of the proposed framework in similarity learning of brain networks by applying it in two classification tasks: (1) Pair classification, and (2) Subject classification.

Methods ABIDE HCP HIV Bipolar
PCA
Siamese GCN
Multi-hop Siamese GCN 0.96 0.02 0.98 0.03 0.77 0.20 0.94 0.07
Table 1: AUC Scores of Pair Classification (mean std).
Figure 3: AUC Scores of Pair Classification.

Pair Classification.

Pair classification refers to the classification of similar pairs (brain networks from the same class) versus dissimilar pairs (brain networks from different classes) based on the similarity learned by the model. This is a very important task in brain connectivity analysis, especially for the brain disorder identification problem when there is very limited number of labeled samples, which is a common scenario in biomedical data mining. If we could build a powerful model that can distinguish similar pairs and dissimilar pairs well, then we could use it for predicting unseen samples by looking into its similarity with the labeled samples. With the Siamese architecture, the proposed model inherently has advantages for this task. For instance, given a training set with samples, we could generate unique pairs of brain networks that could be used as inputs for the Multi-hop Siamese GCN model, which would greatly guarantee the training effectiveness and robustness of the model when applying it for predicting unseen samples. Table 1 and Figure 3 shows the classification AUC of the Multi-hop Siamese GCN and that of the baseline methods on pair classification on the four datasets.

As shown in Table 1, we observe the classification AUC of Multi-hop Siamese GCN is significantly higher than that of the baseline methods. More specifically, our proposed approach achieves an average AUC gain of % compared to PCA, and an average AUC gain of % compared to S-GCN across all datasets. Thus, our proposed method is more accurate and has a lower varaince compared to S-GCN. Among the three methods, the PCA based approach achieved the lowest AUC scores. This is probably due to the fact that PCA learns lower dimensional feature representations directly from the correlation matrix while not considering the structural information of the graph. However, in brain functional connectivity networks, the inner structure usually reflects the collaborative patterns or interactions between different brain regions, which could serve as important features for discriminating brain health status. The Siamese GCN model instead employs the neighborhood structural information of graphs and performs graph convolutions with spectral filtering, which tends to capture localized structural information of graphs. It achieved fairly good results on ABIDE and HCP datasets although not as good as our proposed framework. The superior performance of our proposed Multi-hop Siamese GCN approach indicates that the multi-hop graph convolutions enabled by the high-order structural representation introduced by the random walk stage did help the similarity learning of brain networks. Moreover, the Multi-hop Siamese GCN achieved the best results on all the four datasets, demonstrating its generalizing ability in similarity learning of brain networks.

Subject Classification.

In this experiment, we use the pairwise similarity learned by the model to further classify the subjects with brain disorder versus healthy controls. We evaluate the proposed Multi-hop Siamese GCN model and the baseline Siamese GCN model on the ABIDE and Bipolar datasets. We apply the weighted -nearest neighbour (kNN) classifier [\citeauthoryearHechenbichler and Schliep2004] with the similarity scores we learned for the classification task. The class label of one subject is determined based on a weighted combinations of the class labels of its -nearest neighbors. Here we consider all the neighbors with positive similarity scores in the weighted calculation. Meanwhile, we hope to explore the influence of different loss functions in the subject classification performance. Besides evaluating the two models with the Hinge loss in Equation (5), we also evaluate them with the following constrained variance loss:

(7)

where represents the mean similarity between embeddings belonging to the same class, and represents the mean similarity between embeddings belonging to different classes, while and refer to the variance of pairwise similarity for the same class and different classes, respectively. is the margin between the means of the same-class and different-class similarity distributions, and is the variance threshold. This loss function is proposed by [\citeauthoryearKtena et al.2018]. By this formulation, the variance is only penalised when it exceeds the threshold , which allows the similarity estimates to vary around the means, thus could be used to accommodate the diversity that usually exists in fMRI data due to the varied factors in the acquisition process.

(a) Hinge Loss
(b) Constrained Variance Loss
Figure 4: Subject classification accuracy on ABIDE and Bipolar with two different loss functions

Figure 4 shows the evaluation results of subject classification by the two models with different loss functions. As shown in the Figure, the proposed Multi-hop Siamese GCN model achieves a higher accuracy with both loss functions on both datasets compared to the baseline Siamese GCN model. This implies that the similarity scores learned by the proposed model are more accurate, thus more reliable to be used for further multi-subject brain connectivity analysis. By comparing Figure 4(a) and Figure 4(b), we can find that both models get higher classification accuracy with the similarity scores learned by the constrained variance loss. This could be the benefit from allowing for more diversity across the samples by the constrained variance loss.

3.4 Parameter Analysis

In the proposed Multi-hop Siamese GCN model, there are two sets of parameters. One is the set of parameters for the convolutional networks, and the other set are the parameters for the random walk algorithm. In the experiment, we use GCN layers with features for each. We use the stochastic gradient descent with ADAM algorithm [\citeauthoryearKingma and Ba2014] for the optimization, where we set learning rate to be and use for the polynomial filters in the spectral filtering. We set the dropout rate at the fully connected layer as and use for the regularization parameter. For the constrained variance loss in Equation (7) used in the subject classification task, we set for both datasets. For the parameters in random walk, we employ the grid search in a range of values to find the optimal parameter values. We fix the number of walks to be , and search the value for the walk length from and the value for window size from . For the -nn graph construction stage for refining graph representation, we use of the number of nodes in the brain networks as the value for .

To analyze the influence of the parameters in the random walk process on the similarity learning performance of the proposed model, we perform a parameter sensitivity evaluation for the two key parameters in random walk: the walk/path length and the window size which is also the number of hops considered in the graph convolutions. The bar plots in Figure 5 show the AUC scores of the proposed model with different parameter values for path length and number of hops. From the figure, we observe that the AUC scores vary across different parameter settings. The highest AUC scores on the four datasets tend to come from the cases with longer path length and relatively more hops. For instance, the best result on ABIDE is achieved when path length is and number of hops is , and the best result on HIV is achieved when path length is and number of hops is . The selection of the parameter values also relates to the scale of the brain networks involved. Among the four datasets, HCP has the largest number of nodes. The optimal path length value and number of hops are also the largest among the four datasets. This is reasonable, as the random walks generated on larger graph tend to be longer than those on smaller graphs assuming the two graphs have the same density. To capture the high-order information from the larger graph, we should perform a relatively large number of hops as well. Therefore, it is important to consider the scale and other relevant properties of the brain networks when selecting the parameter values.

(a) ABIDE
(b) Bipolar
(c) HIV
(d) HCP
Figure 5: AUC scores for Pair classification with different values for random walk parameters

4 Related Work

Our work relates to several branches of studies, which include brain network analysis, Siamese networks and graph convolutional networks. Brain network analysis has been an emerging research area, as it yields new insights concerning the understanding of brain function and many neurological disorders [\citeauthoryearLiu et al.2017]. Existing works in brain networks mainly focus on discovering brain network from spatio-temporal voxel-level data or mining from brain networks for neurological analysis [\citeauthoryearMa et al.2016, \citeauthoryearBai et al.2017, \citeauthoryearMa et al.2017a, \citeauthoryearWang et al.2017, \citeauthoryearMa et al.2017b]. For example, in [\citeauthoryearBai et al.2017], an unsupervised matrix tri-factorization method is developed to simultaneously discover nodes and edges of the underlying brain networks in fMRI data. [\citeauthoryearMa et al.2017b] propose a multi-view graph embedding approach which learns a unified network embedding from functional and structural brain networks as well as hubs for brain disorder analysis. In [\citeauthoryearWang et al.2017], a deep model with CNN is proposed to learn non-linear and modular-preserving structures from brain networks for brain disorder diagnosis. Most of these works aim to learn discriminative features from brain networks for the classification or clustering of subjects. However, how to measure the similarity in the graph domain is seldom studied for multi-subject brain network analysis. A general framework for similarity learning on networks is highly demanded for group-contrasting brain network analysis.

Siamese networks were first introduced in [\citeauthoryearBromley et al.1994] to solve image matching problem for signature verification. In recent years, Siamese architecture has drawn much attention from researchers in image recognition [\citeauthoryearChopra, Hadsell, and LeCun2005, \citeauthoryearKoch, Zemel, and Salakhutdinov2015]. In [\citeauthoryearChopra, Hadsell, and LeCun2005], a Siamese framework with CNN is applied to learn similarity metric for face verification, while [\citeauthoryearKoch, Zemel, and Salakhutdinov2015] proposes an approach with Siamese networks to generalize the predictive power of the model to unseen classes and solve one-shot classification problems. Although the Siamese architecture has been widely used in image classification, it is seldom explored in the domain of graph mining. We aim to take advantage of this architecture to learn similarity metric of graphs with Multi-hop GCN, which incorporates high-order graph representation into the learning process.

Graph convolutional network (GCN), first proposed in [\citeauthoryearDefferrard, Bresson, and Vandergheynst2016], is a generalized architecture of convolutional neural network on graphs, where localized spectral filters are defined by Chebyshev polynomials and a recursive formulation is employed for fast filtering operations. In [\citeauthoryearKipf and Welling2016], a renormalization trick is introduced to simplify and speedup computations when applying GCNs for semi-supervised classification. [\citeauthoryearLi, Han, and Wu2018] proposes co-training and self-training approaches to improve the training of GCNs in learning with very few labels. In this work, we apply GCNs in Siamese architecture to obtain important structural features in graphs for similarity learning. As the spectral filters in GCNs are closely related to the graph Laplacian, it is very important to refine the graph representation used in the framework. That motivates us to design the multi-hop GCNs that can capture refined high-order structural information of graphs for a better learning performance. In future work, we aim to explore the impact of attention networks on both pair and subject classification [\citeauthoryearLee et al.2018, \citeauthoryearVeličković et al.2017, \citeauthoryearLee, Rossi, and Kong2018].

5 Conclusion

We present a general framework called Multi-hop Siamese GCN for learning similarity between brain networks. We employ random walks with sliding windows to obtain high-order proximity in graphs and leverage the high-order information in graph convolutional networks for the similarity learning. The multi-hop property allows for multiple options of refining graph representations associated with the graph convolutions, thus the model could be used as a general framework for learning similarity among brain networks from multiple subjects. Extensive experiment results on four real fMRI datasets demonstrate the superior performance of the proposed approach in similarity learning for brain network analysis.

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