Sparsity‐guided multiple functional connectivity patterns for classification of schizophrenia via convolutional network

Abstract The explorations of brain functional connectivity network (FCN) using resting‐state functional magnetic resonance imaging can provide crucial insights into discriminative analysis of neuropsychiatric disorders, such as schizophrenia (SZ). Pearson's correlation (PC) is widely used to construct a densely connected FCN which may overlook some complex interactions of paired regions of interest (ROIs) under confounding effect of other ROIs. Although the method of sparse representation takes into account this issue, it penalizes each edge equally, which often makes the FCN look like a random network. In this paper, we establish a new framework, called convolutional neural network with sparsity‐guided multiple functional connectivity, for SZ classification. The framework consists of two components. (1) The first component constructs a sparse FCN by integrating PC and weighted sparse representation (WSR). The FCN retains the intrinsic correlation between paired ROIs, and eliminates false connection simultaneously, resulting in sparse interactions among multiple ROIs with the confounding effect regressed out. (2) In the second component, we develop a functional connectivity convolution to learn discriminative features for SZ classification from multiple FCNs by mining the joint spatial mapping of FCNs. Finally, an occlusion strategy is employed to explore the contributive regions and connections, to derive the potential biomarkers in identifying associated aberrant connectivity of SZ. The experiments on SZ identification verify the rationality and advantages of our proposed method. This framework also can be used as a diagnostic tool for other neuropsychiatric disorders.


| INTRODUCTION
Schizophrenia (SZ) is a serious chronic disease with unknown etiology (Insel, 2010), and severely affects patient's cognition, emotions (Couture et al., 2006;Tandon et al., 2009), and daily life. It is usually characterized by cognitive distortion, reduced social drive, decreased sociability, and independent living ability. However, SZ diagnosis is complicated due to different symptoms in different individuals, although SZ has led to an international focus on timely diagnosis and earlier intervention.
Resting-state functional magnetic resonance imaging (rs-fMRI) is a non-invasive and highly reproducible brain function imaging method, with no requirement of subjects to perform tasks during the scanning.
This avoids the interference caused by the task execution. The rs-fMRI characterizes local brain spontaneous neural activity by recording blood-oxygenation-level-dependent (BOLD) signals on the time scale (Brown & Eyler, 2006). Brain functional connectivity network (FCN) measures the functional relationship between predefined brain regions of interest (ROIs) based on their BOLD signals, where ROIs are treated as nodes and functional connectivity (FC) between paired nodes refers to the edges (Van Den Heuvel & Pol, 2010). The rs-fMRI-based FCNs have been widely applied to diagnosis of neuropsychiatric disease Zhang et al., 2022). It is crucial to explore a way to construct the FCN that can preserve information contained in rs-fMRI, improve the diagnostic accuracy of SZ.
Pearson's correlation (PC) and sparse representation (SR) are commonly used for constructing FCNs. It has been reported in a previous study that the PC-based methods have relatively high sensitivity for detecting network connections compared with other methods (Smith et al., 2011). PC can measure intrinsic connectivity range from À1 to 1 between paired ROIs, but it cannot measure complex interaction of paired ROIs under the confounding effect of other ROIs.
Meanwhile, the PC-based FCN implicitly assumes that the brain network is densely connected. Emerging evidence has suggested that FCNs constructed considering the influence of multiple ROIs could improve the diagnostic performance of brain diseases (Jie et al., 2016), and the sparsely connected networks are sometimes superior to densely connected ones (Hagmann et al., 2008). The sparse FCNs based on SR have been proposed to characterize the interaction of paired regions under the mutual influence of other ROIs (Lee et al., 2011). In SR, the false connections caused by the lowfrequency spontaneous fluctuations of the BOLD signals and physiological noise are forced to be zero by adding a sparsity prior, and then a sparse FCN can be constructed. Although SR considers the effects of multiple brain regions on the relevance of paired ROIs, the sparse constraint term penalizes each edge equally, which often makes the FCN look like a random network. Compared with PC, the interaction of paired ROIs in the FCN established by SR is weakened, that is, a curtailed amplitude range with SR (Yu et al., 2017. In addition, most brain network analysis methods usually focus on single construction model, which may ignore the complementary that exists in different brain network construction models. This complementary information may be important for brain disease diagnosis. With the enormous development of deep learning algorithms, the FC matrix, associated with good grid properties, reveal great compatibility with deep learning methods (Bi et al., 2020;Malkiel et al., 2021).
For instance, Zhu et al. introduced an autoencoder network with clinically relevant text information for the diagnosis of mild cognitive impairment, which revealed discriminative brain network characteristics (Ju et al., 2017). Kim et al. (2016) proposed a deep neural network for SZ diagnosis and identification of SZ-related abnormal FC, with autoencoder pre-training to initialize weights and l 1 -norm regularization to control weight sparsity. Bi et al. (2020) designed a convolutional neural network (CNN) combined with extreme learning machine to learn brain network regional-connectivity features for Alzheimer's disease diagnosis. Meszlényi et al. (2017) proposed a connectome-CNN combining information from different FCs for mild cognitive impairment diagnosis, and it has been widely used in connectome-based classification tasks. However, these methods are poorly interpretable and ignore the sparsity and spatial properties of functional brain network, which may lead to difficulty in identifying abnormal FC and weak classification performance.
Correlation-based brain networks implicitly assume that ROIs are densely connected, which may not match the real FC properties. Therefore, Li et al. (2021) defined edges by hard thresholding (edges with top 10% connection strength are retained) partial correlations to achieve sparse brain network. Z. Wang et al. (2022) proposed a distribution-guided network thresholding learning method to adaptively generate an FC-specific threshold for each connection in an FC network according to the distribution of connection strength between subject groups for brain disease diagnosis. In another recent study (L. Wang et al., 2021), PC and the k-nearest neighbors graph are used to build sparse connectivity graphs where the connectivity strength of the first k edges of each node is kept, by which the influence of false connections is reduced. However, these sparsification methods ignore the topology in FC networks with different sparsity. Studies have shown that multi-sparse learning may be useful for exploring real FC network representations (Jie et al., 2014).
In this paper, we propose an SZ classification framework based on CNN and sparsity-guided multiple functional connectivity (SMFC). Specifically, a weighted sparse representation (WSR) construction method is introduced to generate multiple FCNs with different sparsities for sparsely guiding the PC matrix. By combining PC and WSR methods, the intrinsic correlation between pairs of ROIs can be preserved and the interactions between multiple ROIs can be considered simultaneously.
Furthermore, by considering the spatial properties of FCN, we introduce an FC convolution (FC-Conv) to learn the functional representation of each sparse FCN, and then combine these feature representations for SZ diagnosis. The contributions are summarized as follows. (1) We propose a new analysis framework (SMFC-Net) integrating PC and WSR methods, as well as a modified CNN for brain disease diagnosis. (2) We obtain better classification performance compared to state-of-the-art (SOTA) methods and discover the important brain regions in classification, which demonstrates the rationality of our method in SZ diagnosis.
(3) The source codes of our method have been released to the public at https://github.com/pancccool/SMFC-Net.  Subjects (n = 147 before screening) are excluded from the study if (1) the average displacement due to head motion during fMRI scanning, as estimated from the realignment parameters, exceeded 0.50 mm (Power et al., 2012); (2) the diagnosis results from the DSM-IV criteria are unrelated to SZ (n = 4 subjects); (3) the data are disenrolled (n = 2 subjects); and (4) the data acquisition process is incomplete (n = 1 subject). The final remaining subjects included 57 SZs and 68 HCs. The demographic information of the data set is shown in Table 1.

| Overview of the framework
We propose to construct a new framework to establish the sparsityguided multiple FCs, which combines the advantages of both PC and WSR. Figure 1 shows the schematic illustration of our proposed SMFC-Net, consisting of two components, that is, (1) one component is the construction of FCNs and (2) another one is a CNN. Specifically, we first compute the PC coefficient matrix P to characterize the connectivity strength using the resting-state BOLD signal. Then, we construct d sparsity matrices (i.e., {W λk }) based on WSR with weighted sparsity constraint term (Yu et al., 2017) to represent different network sparse topologies (d = 10 in our work), where λ k represents the sparse prior parameters. W λk are first binarized (0/1) and then applied to P to obtain the sparsity-guided FC matrix M λk as shown in 2.3 | Convolutional neural network with sparsityguided multiple functional connectivity 2.3.1 | PC matrix PC directly measures FC of ROIs by calculating the correlation coefficient of paired ROIs, ranging from À1 to 1. The PC coefficient between two brain regions is calculated by where normalized mean time series of the ⅈth and jth ROIs, respectively. L = 140 is the length of the time series. The whole brain PC matrix is

| Network based on WSR
In our previous research (Yu et al., 2017), the WSR is able to characterize interactions of paired ROIs with the interference of other ROIs in the network by combining sparsity and pairwise similarity.
The WSR is depicted by a weighted l 1 -norm, which is formulated by where w i ¼ W 1i ,W 2i , ÁÁÁ, W Ni ½ T denotes a column vector composed of the connecting edges between the ith ROI and all the others.
nary when expressing the BOLD signal x i of the ith brain region. The ith column of the dictionary is set to 0 to avoid trivial solutions. We as a penalty weight vector corresponding to w i , and as the element-wise multiplication. The penalty weight C ji is defined as an inverse proportion function of the pair-wise correlation P ji , that is, C ji ¼ exp ÀP 2 ji =σ , where σ denotes the parameter to adjust the attenuation speed of the corresponding connectivitystrength weight empirically set as 0.2 to be same as our previous work on the identification of mild cognitive impairment (Yu et al., 2017. Thus, a larger correlation P ji will result in a smaller penalty C ji and also a weaker constraint on the connectivity W ji . In contrast, a larger penalty C ji will push W ji to approach 0. For whole-brain FC network construction, Equation (2) is rewritten as its equivalent matrix form: (3), the alternating direction method of multiplier (Boyd et al., 2011) is adopted to calculate the connectivity network W. The detailed optimization process of Equation (3) can refer to . λ is a regularization parameter, related to the sparsity prior of WSR. Setting the sparsity prior λ with different values in range of 2 À4 ,2 À3 , ÁÁÁ,2 5 According to Equation (3), WSR (penalizing the edges with different weights) is superior to SR which penalizes edges with a constant weight (i.e., 1). However, the edges corresponding to strong correlation are still penalized with the penalty weight, which makes the difference between strong and weak connectivity edges in the FC insignificant (i.e., the range of correlation is curtailed).
The framework of our proposed SMFC-Net for SZ classification. (a) The construction process of FCNs. The time series are extracted from N ROIs, and a Pearson's correlation matrix P is calculated from the time series. W λk is a sparse matrix calculated based on weighted sparse representation, where λ is the regularization parameter and its range is 2 À4 ,2 À3 , ÁÁÁ,2 5 h i . M λk is the sparsity-guided FC matrix. In part (a), the red arrow indicates the combination of P and W λk into the sparsity-guided multiple FCs matrix M λk through sparse guidance, and the black dotted arrow indicates the construction process of W λk . (b) The convolutional neural network. Each FC-Conv contains two convolutional layers, that is, Conv1 (regional connectivity convolution layer) and Conv2 (spatial integration convolution layer). Here, the kernel sizes in two layers are 1 Â N and N Â 1 (with the corresponding channel numbers of H and V), respectively. "Concatenation" denotes that we connect the output (i.e., vector 1 Â 1 Â V) of all FC-Convs to obtain 1 Â 1 Â 10V features. Furthermore, the two fully connected layers (with their numbers of units as D 1 and D 2 , respectively) and a softmax layer are used for SZ classification. FC-Conv, functional connectivity convolution; FCN, functional connectivity network; ROI, region of interest; SMFC-Net, convolutional neural network with sparsity-guided multiple functional connectivity; SZ, schizophrenia.
2.3.3 | The construction of sparsity-guided multiple functional connectivity network To characterize connectivity between paired ROIs influenced by the other ROIs, we propose to construct the sparsity-guided multiple FCs based on PC and WSR. Specifically, we use PC to represent the connectivity strength, and WSR to represent the network sparse topology. Since the W λk constructed based on WSR is an asymmetric matrix, we adopt a strategy in Elhamifar and Vidal (2013) to symmetrize W λk . Then, a binarization (0/1) operation is performed on W λk , by which the non-zero elements in W λk are set to 1, and the zero elements remain 0. The binarized format of W λk is denoted as G λk ℝ NÂN and the sparsity-guided multiple FCs are calculated by the following dot product operation on P and G λ k : The M λk keeps strong linear correlation as in P, and alleviates the impact of the weakened connectivity strength caused by WSR. Applying G λk to P, the resulting FCN can both retain strong correlation for non-zero interactions and eliminate false connections caused by the influence of other brain regions apart from the paired ROIs. By introducing a sparsity prior λ in WSR, the sparse FCN M λk can not only indicate the existence of crucial edges in P, but also remove the false connections effectively.

| Architecture of CNN
To explore the potential discriminative connectivity for SZ classification, the sparsity-guided multiple FCs are used as the input of next step as shown in Figure 1b. Each sparse FCN corresponds to a two-layer CNN, that is, FC-Conv. For M λ k , each row/column in the FCN represents the correlation between a specific ROI and all the other ROIs. To find the latent high-order spatial representation in the FCNs, we first employ a regional connectivity convolution kernel to learn the spatial mapping between a specific ROI and other ROIs by weighting and summing each row in the first layer. Specifically, we set the size of the regional connectivity convolution kernel as 1 Â N Â H, where H is the number of channels, and then the output of the first layer is an N Â 1 Â H tensor.
To integrate the spatial mapping learned by the regional connectivity convolution, we introduce a spatial integration convolution kernel with the size of N Â 1 Â V in the second layer, and the output is the predictive feature with a reduced dimension learned from each FCN, that is, a 1 Â 1 Â V flattened tensor. These predictive features from multiple branches are transformed into a 1 Â 1 Â d Â V ð Þvector, and input into two fully connected layers (with D 1 and D 2 neurons, respectively), and a softmax unit is used for SZ classification.

| Implementation of SMFC-net
The proposed architecture is implemented using python3 based on ten-sorflow2 and trained on a single GPU (NVIDIA GeForce RTX 3070) with 8GB of memory. In each FC-Conv, the initialization of weights for two convolutional layers is "he_normal" (He et al., 2015), and the numbers of channels are set as H ¼ 64 and V ¼ 32, respectively. Each convolutional layer is followed by l 2 -norm of convolution kernel weight matrix (l 2 regular term coefficient is 0.00001). "Concatenation" and the first fully connected layer are followed by 0.2 dropout (Srivastava et al., 2014).
The numbers of neurons in the two fully connected layers are D 1 ¼ 128 and D 2 ¼ 64, respectively. The number of neurons in the softmax layer is set to 2 for SZ classification. The Adam optimizer is used for training (Kingma & Ba, 2014), and the number of epochs, batch size and learning rate are empirically set as 200, 20, and 0.001, respectively. The SMFC-Net is trained on the training data to minimize the cross-entropy cost.

| Experimental settings
We employ the fivefold cross-validation (CV) to evaluate our pro- is used to integrate the prediction results of these base classifiers to obtain the final prediction result. The above process is repeated 10 times independently for estimating the final performance.
We employ seven metrics to evaluate the performance of the method, including accuracy (ACC), sensitivity (SEN), specificity (SPE), balanced accuracy (BAC), receiver operating characteristic (ROC), area under the curve (AUC), and F 1 -score (F 1 ). The formulas of ACC, SEN, SPE, BAC, and F 1 are as follows:

| Method comparison
The proposed SMFC-Net is compared with five conventional machine/deep learning methods, one advanced graph neural network (GNN) method as well as some variants of the SMFC-Net.

| Classification performance
The SZ classification performance using different methods is summarized in Tables 2 and 3, and the ROC curves are shown in Figure 2, where SZs are treated as the positive class.
According to the results, one can have the following observa- results of the existing research (Meszlényi et al., 2017). Compared with WSR-CNN, PC-CNN achieves relatively better performance, indicating that the weak connection had limited contribution to pattern recognition in deep learning. Second, compared with HTFC-Net, the performance confirms that our proposed sparsity-guided multiple FCs retain more discriminative FC, and eliminates more false connections in the FCNs. Finally, our SMFC-Net achieves better performance than its variants, that is, SSFC-Net (λ 2 ), SSFC-Net (λ 4 ), SSFC-Net (λ 6 ), SSFC-Net (λ 8 ), and SSFC-Net (λ 10 ), which implies that our strategy of employing multiple sparse learning can significantly contribute to the improvement of classification performance.

| Comparison with sota methods
We further compare our method with several SOTA studies using the COBRE data set. The results are listed in Table 4, where the data T A B L E 3 The comparison of performance between the proposed SMFC-Net and variant methods in SZ classification (mean [std]%). while previous methods only focus on one aspect of these information or ignore the spatial information in FC networks.

| Important functional connectivities in classification
The occlusion method (Zeiler & Fergus, 2014) is applied to explore the important FC in SZ classification. We set the FC (i.e., element) existing in brain network (i.e., M λk matrix) to zero in turn, and send it to the trained SMFC-Net model to get a new classification accuracy. By comparing the classification accuracy of SMFC-Net, the importance of these FC is obtained according to the degradation of classification performance. Specifically, since M λ k is symmetrical matrix, that is, M λk ij is equal to M λk ji , we occlude two elements (i.e., the connections M λk ij and M λk ji ) of the input matrix M λk by replacing them with a zeroocclusion mask. The zero-occlusion mask moves in {M λ k } at the same time, and the occluded {M λk } obtained by each movement is sent to the trained SMFC-Net to obtain a new classification accuracy.
=2 classification accuracies can be obtained, and they are subtracted from the accuracies of SMFC-Net (78.16%) separately.
The degradation of classification accuracies is treated as the contribution of each FC.
The top 10 important FCs in classification are shown in Figure 3.
The names and abbreviations of the corresponding ROIs are listed in Table 5.
From Figure 3 and  Li et al., 2012), these regions have also been reported to be highly correlated with SZ progression.
Among these connections, SZ patients have greater thalamic connectivity with multiple sensory-motor regions which is consistent with (Ferri et al., 2018). These results indicate that our method is effective in identifying SZ-related FC.
F I G U R E 3 Top 10 important functional connectivity contributing the most in SZ classification. SZ, schizophrenia.

| Important brain regions in classification
We also apply the occlusion method to identify the important brain regions for SZ classification. Specifically, we perturb all the connections with a specific ROI in the input matrix M λk by replacing them with an occlusion mask, typically a zero row/column vector. The mask moves across the matrices one ROI at a time. The degradation in accuracy is normalized to evaluate the contribution of each ROI. The contribution of brain regions in each independent experiment is as follows: where D n denotes the exponential form of degradation in accuracy, n 1,2,ÁÁÁ, N ½ , ACC our denotes the ACC of SMFC-Net, ACC masked denotes the ACC obtained by feeding the occlusion masked input M λk into the trained SMFC-Net, and C n denotes the normalized D n to indicate the contribution of the nth ROI. (PoCG) left. The importance of these brain regions in SZ diagnosis has been reported in previous studies (Andreasen, 1997;Hoptman et al., 2010;Ichimiya et al., 2001;Onitsuka et al., 2004;Taylor et al., 2002;van Erp et al., 2016). Specifically, compared with HC, the SZ patients showed a decreased activation in SMA (Schröder et al., 1995), and a reduced gray matter density in both THA and PoCG (Glahn et al., 2008). The relatively high contribution values of these brain regions are also demonstrated in Figures 4b and 5.

| DISCUSSION
Integrative analysis using multiple types of measurement for FC, such as PC and WSR, can take advantage of their complementary information and thus help construct sparse FCN containing more complex brain region interactions. In this paper, we propose sparsity-guided multiple FCs, by integrating PC and WSR. This strategy not only retains the edge weights between paired ROIs, but also indicates the F I G U R E 4 (a) The specific contributions of different ROIs in fivefold cross-validation (CV) for 10 repetitive experiments. The vertical axis represents the number of CV, and the light/dark colors denote smaller/larger contribution of each ROI for classification. (b) The overall contributions of all cross-validated ROIs in SZ classification. The vertical axis represents the overall contribution value.
The horizontal axis represents the ROIs. ROI, region of interest; SZ, schizophrenia.
existence of edges in FCNs. This method can effectively and efficiently remove internal redundancy for constructing connectivity and generate multiple different sparse FCNs, which shows significantly advantage compared with previous studies that focus only on a single construction method (e.g., only using PC or WSR). Different from hard thresholding strategy, the sparsity-guided method can more effectively and accurately eliminate false connections from the dense whole-brain FCNs, by which FCNs with different levels of sparsity are generated.
For SZ diagnosis, we establish an SMFC-Net framework to extract discriminative features from sparsity-guided multiple FCs.
SMFC-Net can capture the high-order spatial pattern, meanwhile considering the information complementarity of different sparsity FCNs.
An occlusion strategy is applied to investigate the important FCs and brain regions for SZ classification, which can be regarded as a potential biomarker for identifying SZ-associated aberrant connectivity.
Interestingly, most of the connections listed in Table 5 focus on SZpathology-related brain regions that have been reported in previous studies as illustrated in previous sections.
To the best of our knowledge, our proposed method is the first attempt to integrate complementary information PC and WSR for constructing FCNs, and to build a CNN framework for analyzing those FCNs. The experimental results show that our proposed method outperforms several other FCN-based methods, and demonstrates that our proposed method may provide important insights in revealing the underlying interactions of brain regions.
Although our proposed method has achieved great performance, it is still limited by the following several aspects. (1) The size of the data set uses in this study is relatively small, that is, only 125 subjects are used to evaluate the performance of the proposed method.
Besides, the number of samples in different categories is also not balanced. In the future, we will evaluate our method on a larger and more balanced data set, that is, the autism spectrum disorder ASD data (Di Martino et al., 2014) from the Autism Brain Imaging Data Exchange data set.
(2) Our work uses the static information in the BOLD signal to identify the disease. While, studies have reported that the dynamic information in the BOLD signals is conducive to improving the classification performance (M. Wang et al., 2019). Combining a variety of latent representations in BOLD to diagnose diseases will be our future work.
(3) This work uses CNN architecture to identify the FC. The GNN has outstanding performance in recognizing patterns in FCNs (Huang et al., 2020b). In the future work, we will develop the framework based on the GNN to mine FCN, such that the classification performance can be potentially improved.
F I G U R E 5 Visualization of contribution level of each brain region in the SZ classification. The color bar represents the contribution from small (blue) to large (red). SZ, schizophrenia.

| CONCLUSION
This paper introduces a novel SZ classification framework (SMFC-Net) based on fMRI data. Our proposed method integrates PC which measures the statistical relationship between paired ROIs/nodes (aka, edge weight in FCN) and WSR which determines the existence of edges to construct sparsity-guided multiple FCs. The FC-Conv is applied to capture the high-order joint spatial mapping between each central ROI and all other ROIs, and the learned embedding features from multiple structures of FCNs are combined for SZ classification.
The experimental results from the COBRE database demonstrate the promising performance of our proposed method compared to the SOTA.