GEMINI: GRAPH ESTIMATION WITH MATRIX VARIATE NORMAL INSTANCES

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 conditio...

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Bibliographic Details
Published in:The Annals of statistics 2014-04, Vol.42 (2), p.532-562
Main Author: Zhou, Shuheng
Format: Article
Language:English
Subjects:
62F12
62F30
Correlations
Covariance
covariance estimation
Covariance matrices
Estimating techniques
Estimation methods
Estimators
Gaussian distributions
Graph theory
graphical Lasso
Graphical model selection
inverse covariance estimation
Kronecker product
Machine learning
Matrices
Matrix
matrix variate normal distribution
Normal distribution
Parameter estimation
Perceptron convergence procedure
Simulation
Studies
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/13-AOS1187