Covariance estimation in decomposable Gaussian graphical models

Ami Wiesel*, Yonina C. Eldar, Alfred O. Hero

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Scopus citations

Abstract

Graphical models are a framework for representing and exploiting prior conditional independence structures within distributions using graphs. In the Gaussian case, these models are directly related to the sparsity of the inverse covariance (concentration) matrix and allow for improved covariance estimation with lower computational complexity. We consider concentration estimation with the mean-squared error (MSE) as the objective, in a special type of model known as decomposable. This model includes, for example, the well known banded structure and other cases encountered in practice. Our first contribution is the derivation and analysis of the minimum variance unbiased estimator (MVUE) in decomposable graphical models. We provide a simple closed form solution to the MVUE and compare it with the classical maximum likelihood estimator (MLE) in terms of performance and complexity. Next, we extend the celebrated Stein's unbiased risk estimate (SURE) to graphical models. Using SURE, we prove that the MSE of the MVUE is always smaller or equal to that of the biased MLE, and that the MVUE itself is dominated by other approaches. In addition, we propose the use of SURE as a constructive mechanism for deriving new covariance estimators. Similarly to the classical MLE, all of our proposed estimators have simple closed form solutions but result in a significant reduction in MSE.

Original languageAmerican English
Pages (from-to)1482-1492
Number of pages11
JournalIEEE Transactions on Signal Processing
Volume58
Issue number3 PART 2
DOIs
StatePublished - Mar 2010
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received March 06, 2009; accepted September 28, 2009. First published November 24, 2009; current version published February 10, 2010. The work of A. Wiesel was supported by a Marie Curie Outgoing International Fellowship within the 7th European Community Framework Programme. This work was supported in part by AFOSR MURI Grant FA9550-06-1-0324. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Deniz Erdogmus.

Keywords

  • Covariance estimation
  • Graphical models
  • Minimum variance unbiased estimation

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