Chen Zu^{1}, Yue Gao^{2}, Brent Munsell^{3}, Minjeong Kim^{1}, Ziwen Peng^{4}, Yingying Zhu^{1}, Wei Gao^{5}, Daoqiang Zhang^{6}, Dinggang Shen^{1}, and Guorong Wu^{1}

Most brain network connectivity models consider correlations between discrete-time series signals that only connect two brain regions. Here we propose a method to explore subnetwork biomarkers that are significantly distinguishable between two clinical cohorts. We construct a hypergraph by exhaustively inspecting all possible subnetworks for all subjects. The objective function of hypergraph learning is to jointly optimize the weights for all hyperedges. We deploy our method to find high order childhood autism biomarkers from rs-fMRI images. Promising results have been obtained from comprehensive evaluation on the discriminative power in diagnosis of Autism.

Figure 1 illustrates the intuition behind our proposed learning-based method. We assume there are three subjects in one cohort and two subjects in another cohort. Only two possible subnetworks are under investigation. Our method aims to find out the biomarkers by inspecting the performance of each hyperedge in separating subjects from two groups. We use hypergraph learning technique to estimate the likelihood for each subject , which is driven by (a) the minimization of discrepancies between ground truth label vector $$$y$$$ and the estimated likelihood vector $$$f=[f_1,f_2,\dots,f_N]^T$$$, and (b) the consistency of clinical labels within each hyperedge. The consistency requirement can be defined as:

$$\Omega_f(W)=\sum_{\theta=1}^{\Theta}\sum_{n,n'=1}^ N=\frac{w_{\theta}h(n,\theta)h(n',\theta)}{\delta(\theta)}(\frac{f_n}{\sqrt{d(n)}}-\frac{f_{n'}}{\sqrt{d(n')}})^2$$

The definitions of notations can be found in [1]. The regulation term $$$\Omega_f(W)$$$ penalizes the label discrepancy by encouraging the difference between the normalized likelihoods $$$ \frac{f_n}{\sqrt{d(n)}}$$$ and $$$\frac{f_{n'}}{\sqrt{d(n')}}$$$ to be as small as possible if $$$v_n$$$ and $$$v_{n'}$$$ are in the same hyperedge $$$e_{\theta}$$$. It is clear that the regularization term $$$\Omega_f(W)$$$ is a function of both $$$W$$$ and $$$f$$$. In order to avoid overfitting, we use Frobenius norm on the weighting matrix $$$W$$$. Therefore, the objective function to look for high order connectome patterns is:

$$\min_{W,f}\quad \Omega_f(W)+\lambda\lVert y-f\rVert_2^2+\mu\lVert W\rVert_F^2$$

[1] Zhou D, Huang J, Schölkopf B. Learning with hypergraphs: Clustering, classification, and embedding. Advances in neural information processing systems. 2006: 1601-1608.

[2] Di Martino, A., Yan, C.G., Li, Q., Denio, E., Castellanos, F.X., Alaerts, K., Anderson, J.S., Assaf, M., Bookheimer, S.Y., Dapretto, M., et al.: The autism brain imaging data exchange: towards a large-scale evaluation of the intrinsic brain architecture in autism. Molecular psychiatry 19(6), 659-667 (2014)

[3] Minshew, N.J., Williams, D.L.: The new neurobiology of autism: cortex, connectivity, and neuronal organization. Archives of Neurology 64(7), 945-950 (2007)

[4] Cortes, C., Vapnik, V.: Support-vector networks. Machine Learning 20(3), 273-297 (1995)

[5] Tao, D., Li, X., Wu, X., Hu, W., Maybank, S.J.: Supervised tensor learning. Knowledge and Information Systems 13(1), 1-42 (2007)

Figure 1. The overview of our learning-based method to
discover high order brain connectome patterns by hypergraph inference.

Figure 2. The top 10 selected subnetworks (white triangle cliques)
where the functional connectivity flow running inside has significant
difference between ASD and TC cohorts.

Figure 3. Classification performance of four different
classification methods.