High dimensional convergence rates for sparse precision estimators for matrix-variate data

TL;DR

This paper analyzes the convergence rates of two estimators for row and column covariance matrices in high-dimensional matrix-variate data, correcting previous inaccuracies and introducing a novel proof approach.

Contribution

It provides the first correct high-dimensional convergence rate analysis for the SMGM estimator and proposes a new method to establish these rates.

Findings

Corrected previous convergence rate results for SMGM estimator

Established convergence rates for a heuristic covariance estimator

Highlighted errors in prior high-dimensional analyses

Abstract

In several applications, the underlying structure of the data allows for the samples to be organized into a matrix variate form. In such settings, the underlying row and column covariance matrices are fundamental quantities of interest. We focus our attention on two popular estimators that have been proposed in the literature: a penalized sparse estimator called SMGM and a heuristic sample covariance estimator. We establish convergence rates for these estimators in relevant high-dimensional settings, where the row and column dimensions of the matrix are allowed to increase with the sample size. We show that high-dimensional convergence rate analyses for the SMGM estimator in previous literature are incorrect. We discuss the critical errors in these proofs, and present a different and novel approach.