Tensor Graphical Model: Non-Convex Optimization and Statistical Inference

We consider estimation and inference of precision matrices in sparse tensor graphical models. To facilitate study, data is assumed to follow a tensor normal distribution whose covariance has a Kronecker product structure. A critical challenge in both estimation and inference of this model is that its penalized maximum likelihood estimation involves minimizing a non-convex objective function. To address it, this paper makes two contributions: (i) In spite of the non-convexity of estimation, we prove that an alternating minimization algorithm attains an estimator with the optimal statistical rate of convergence. Notably, such an estimator achieves estimation consistency with only one tensor sample, which was not observed in previous works. (ii) A novel de-biased statistical inference procedure with false discovery rate (FDR) control is proposed for testing true support of sparse precision matrices. Asymptotic normality of test statistic and consistency of FDR control are justified. These theoretical results are backed up by thorough simulations. We further illustrate the efficacy of our models through two real applications: neuroimaging studies of Autism spectrum disorder and users' clicking behavior analysis in online advertising. The proposed methods are encoded into a publicly available R package Tlasso.