Identifying disease-related subnetwork connectome biomarkers by sparse hypergraph learning

Abstract

The functional brain network has gained increased attention in the neuroscience community because of its ability to reveal the underlying architecture of human brain. In general, majority work of functional network connectivity is built based on the correlations between discrete-time-series signals that link only two different brain regions. However, these simple region-to-region connectivity models do not capture complex connectivity patterns between three or more brain regions that form a connectivity subnetwork, or subnetwork for short. To overcome this current limitation, a hypergraph learning-based method is proposed to identify subnetwork differences between two different cohorts. To achieve our goal, a hypergraph is constructed, where each vertex represents a subject and also a hyperedge encodes a subnetwork with similar functional connectivity patterns between different subjects. Unlike previous learning-based methods, our approach is designed to jointly optimize the weights for all hyperedges such that the learned representation is in consensus with the distribution of phenotype data, i.e. clinical labels. In order to suppress the spurious subnetwork biomarkers, we further enforce a sparsity constraint on the hyperedge weights, where a larger hyperedge weight indicates the subnetwork with the capability of identifying the disorder condition. We apply our hypergraph learning-based method to identify subnetwork biomarkers in Autism Spectrum Disorder (ASD) and Attention Deficit Hyperactivity Disorder (ADHD). A comprehensive quantitative and qualitative analysis is performed, and the results show that our approach can correctly classify ASD and ADHD subjects from normal controls with 87.65 and 65.08% accuracies, respectively.

Appendix

Support Vector Machine:

Given a data set $D={\left\{{\boldsymbol{x}}_i,{y}_i\right\}}_{i=1}^n$ of labeled samples, where y_i ∈ {−1, +1}. SVM wants to find the optimal hyperplane which can separate the data and minimize the generalization error at the same time. The optimization problem of SVM can be defined as follows:

$$ {\displaystyle \begin{array}{c}\mathrm{mi}{\mathrm{n}}_{\boldsymbol{w},b,\boldsymbol{\xi}}\frac{1}{2}{\boldsymbol{w}}^T\boldsymbol{w}+C\sum \limits_i^n{\xi}_i\kern1em \\ {}\begin{array}{c}\mathrm{s}.\mathrm{t}.{y}_i\left({\boldsymbol{w}}^T{\boldsymbol{x}}_i+b\right)\ge 1-{\xi}_i\\ {}{\xi}_i\ge 0,i=1,2,\dots, n\end{array}\end{array}} $$

(11)

where w is a vector orthogonal to the hyperplane. Equation (11) is a constrained optimization problem and can be solved by using quadratic programming tsechniques.

When a new testing data point coming, a label is assigned to the new sample via the following decision function:

$$ g\left(\boldsymbol{x}\right)=\operatorname{sign}\left({\boldsymbol{w}}^T\boldsymbol{x}+b\right) $$

(12)

Support Tensor Machine.

Given a set of training samples {X_i, y_i}, i = 1, 2, …, n, where X_i is the data point in order-2 tensor space, ${\boldsymbol{X}}_i\in {\mathbb{R}}^{d_1}\bigotimes {\mathbb{R}}^{d_2}$ and y_i ∈ {−1, +1} is the label of X_i. The goal of STM is to find a tensor classifier f(X) = u^TXp + b such that the two classes can be separated with maximum margin. Thus, the optimization problem of STM is as follows:

$$ {\displaystyle \begin{array}{l}{\min}_{u,p,b,\xi}\frac{1}{2}{\left\Vert {\boldsymbol{u}\boldsymbol{p}}^T\right\Vert}^2+c\sum \limits_i^n\xi i\\ {}\mathrm{s}.\mathrm{t}.{\mathrm{y}}_i\left({\boldsymbol{u}}^T{\boldsymbol{X}}_i\boldsymbol{p}+b\right)\ge 1-\xi i\\ {}\xi i\ge 0,i=1,2,\dots, n\end{array}} $$

(13)

As we can see from Eq. (13), STM is a tensor generation of SVM. The algorithm to solve STM is stated below:

• 1. Initialization: Let u = (1, …, 1)^T.

• 2. Calculating p: Let ${\boldsymbol{x}}_i={\boldsymbol{X}}_i^T\boldsymbol{u}$ and β₁ = ‖u‖², p can be computed by solving the following problem:

$$ {\displaystyle \begin{array}{l}{\min}_{p,b,\xi}\frac{1}{2}{\beta}_1{\boldsymbol{p}}^T\boldsymbol{p}+\boldsymbol{c}\sum \limits_i^n{\xi}_i\\ {}\mathrm{s}.\mathrm{t}.{\mathrm{y}}_{\mathrm{i}}\left({p}^T{x}_i+b\right)\ge 1-\xi i\\ {}\xi i\ge 0,1=1,2,\dots, n\end{array}} $$

(14)

It is worth noting that problem (14) is the same as objective function (11) of SVM. Thus, the standard optimization approach for SVM can be adopted for Eq. (14).

• 3. Calculating u:When p is solved, let $\overset{\sim }{{\boldsymbol{x}}_i}={\boldsymbol{X}}_i\boldsymbol{p}$ and β₂ = ‖p‖². u can be computed by solving the following problem:

$$ {\displaystyle \begin{array}{l}{\min}_{u,b,\xi}\frac{1}{2}{\beta}_2{\boldsymbol{u}}^Tu+c\sum \limits_i^n{\xi}_i\\ {}s.t.{y}_i\left({\boldsymbol{u}}^T{x}_i+b\right)\ge 1-{\xi}_i\\ {}{\xi}_i\ge 0,1=1,2,\dots, n\end{array}} $$

(15)

As above, the standard optimization method for SVM can also be used to solve Eq. (15).

• 4. Iteratively computing u and p: u and p can be iteratively calculated by step 2 and 3 until a convergence attained.