Gemini: Graph estimation with matrix variate normal instances

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.

Published in at http://dx.doi.org/10.1214/13-AOS1187 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• One-step sparse estimates in nonconcave penalized likelihood models
• Sparse permutation invariant covariance estimation
• Network exploration via the adaptive LASSO and SCAD penalties
• Convergence Properties of Kronecker Graphical Lasso Algorithms
• Are a set of microarrays independent of each other?

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