Gemini: Graph estimation with matrix variate normal instances

TL;DR

This paper introduces new methods for estimating graphical structures and covariance matrices from matrix variate normal data, demonstrating theoretical guarantees and practical effectiveness with simulations.

Contribution

It develops novel estimation techniques for graphical models in matrix variate distributions, extending to replicate data and providing convergence rates and consistency results.

Findings

Successful recovery of graphical structures from a single matrix

Improved estimation accuracy with replicates

Theoretical convergence rates established

Abstract

Undirected graphs can be used to describe matrix variate distributions. In this paper, we develop new methods for estimating the graphical structures and underlying parameters, namely, the row and column covariance and inverse covariance matrices from the matrix variate data. Under sparsity conditions, we show that one is able to recover the graphs and covariance matrices with a single random matrix from the matrix variate normal distribution. Our method extends, with suitable adaptation, to the general setting where replicates are available. We establish consistency and obtain the rates of convergence in the operator and the Frobenius norm. We show that having replicates will allow one to estimate more complicated graphical structures and achieve faster rates of convergence. We provide simulation evidence showing that we can recover graphical structures as well as estimating the precision matrices, as predicted by theory.