Unified framework to regularized covariance estimation in scaled Gaussian models

Ami Wiesel*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

80 Scopus citations

Abstract

We consider regularized covariance estimation in scaled Gaussian settings, e.g., elliptical distributions, compound-Gaussian processes and spherically invariant random vectors. Asymptotically in the number of samples, the classical maximum likelihood (ML) estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex. We propose a unified framework for regularizing this estimate in order to improve its finite sample performance. Our approach is based on the discovery of hidden convexity within the ML objective. We begin by restricting the attention to diagonal covariance matrices. Using a simple change of variables, we transform the problem into a convex optimization that can be efficiently solved. We then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices. We regularize the problem using appropriately convex penalties. These allow for shrinkage towards the identity matrix, shrinkage towards a diagonal matrix, shrinkage towards a given positive definite matrix, and regularization of the condition number. We demonstrate the advantages of these estimators using numerical simulations.

Original languageAmerican English
Article number6035802
Pages (from-to)29-38
Number of pages10
JournalIEEE Transactions on Signal Processing
Volume60
Issue number1
DOIs
StatePublished - Jan 2012

Bibliographical note

Funding Information:
Manuscript received April 28, 2011; revised August 11, 2011; accepted September 12, 2011. Date of publication October 06, 2011; date of current version December 16, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Alfred Hanssen. This work was partially supported by Israel Science Foundation Grant No. 786/11.

Keywords

  • Covariance estimation
  • hidden convexity
  • optimization on manifolds
  • regularization
  • robust statistics

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