Generalized Connectivity Matrix Response Regression with Applications in Brain Connectivity Studies

Multiple-subject network data are fast emerging in recent years, where a separate connectivity matrix is measured over a common set of nodes for each individual subject, along with subject covariates information. In this article, we propose a new generalized matrix response regression model, where the observed network is treated as a matrix-valued response and the subject covariates as predictors. The new model characterizes the population-level connectivity pattern through a low-rank intercept matrix, and the effect of subject covariates through a sparse slope tensor. We develop an efficient alternating gradient descent algorithm for parameter estimation, and establish the nonasymptotic error bound for the actual estimator from the algorithm, which quantifies the interplay between the computational and statistical errors. We further show the strong consistency for graph community recovery, as well as the edge selection consistency. We demonstrate the efficacy of our method through simulations and two brain connectivity studies. Supplementary materials for this article are available online.近年来，多被试网络数据（multiple-subject network data）快速兴起，此类数据针对每名独立被试，在一组公共节点上分别测量其连通性矩阵（connectivity matrix），同时附带被试协变量（subject covariates）信息。本文提出一种全新的广义矩阵响应回归模型（generalized matrix response regression model）：将观测到的网络视为矩阵值响应（matrix-valued response），将被试协变量作为预测变量（predictors）。该新模型通过低秩截距矩阵（low-rank intercept matrix）刻画总体水平连通模式，通过稀疏斜率张量（sparse slope tensor）表征被试协变量的效应。我们开发了一种高效的交替梯度下降算法（alternating gradient descent algorithm）用于参数估计（parameter estimation），并为该算法得到的实际估计量推导了非渐近误差界（nonasymptotic error bound），该界可量化计算误差与统计误差之间的相互影响。我们进一步证明了图社区恢复（graph community recovery）的强相合性（strong consistency）以及边选择相合性（edge selection consistency）。我们通过模拟实验（simulations）与两项脑连通性研究（brain connectivity studies）验证了所提方法的有效性。本文的补充材料可在线获取。