Tensor Graphical Model: Non-convex Optimization and Statistical Inference

Abstract

We consider the estimation and inference of graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. A critical challenge in the estimation and inference of this model is the fact 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 this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with an optimal statistical rate of convergence. (ii) We propose a de-biased statistical inference procedure for testing hypotheses on the true support of the sparse precision matrices, and employ it for testing a growing number of hypothesis with false discovery rate (FDR) control. The asymptotic normality of our test statistic and the consistency of FDR control procedure are established. Our theoretical results are backed up by thorough numerical studies and our real applications on neuroimaging studies of Autism spectrum disorder and users' advertising click analysis bring new scientific findings and business insights. The proposed methods are encoded into a publicly available R package Tlasso.

arXiv comment63 pages

References in corpus (20)

• Nearly unbiased variable selection under minimax concave penalty
• Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination
• The huge Package for High-dimensional Undirected Graph Estimation in R
• Network exploration via the adaptive LASSO and SCAD penalties
• Asymptotic normality and optimalities in estimation of large Gaussian graphical models
• Provable Tensor Factorization with Missing Data
• Global Optimality in Tensor Factorization, Deep Learning, and Beyond
• Simple, Efficient, and Neural Algorithms for Sparse Coding
• Convergence Properties of Kronecker Graphical Lasso Algorithms
• New Algorithms for Learning Incoherent and Overcomplete Dictionaries
• Alternating Minimization for Mixed Linear Regression
• Fast matrix completion without the condition number
• Computational Limits for Matrix Completion
• More Algorithms for Provable Dictionary Learning
• Complete Dictionary Recovery over the Sphere
• Thresholded Multiscale Gaussian Processes with Application to Bayesian Feature Selection for Massive Neuroimaging Data
• Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models
• Support recovery without incoherence: A case for nonconvex regularization
• Hypothesis Testing of Matrix Graph Model with Application to Brain Connectivity Analysis
• Graph Estimation for Matrix-variate Gaussian Data