Robust shrinkage estimation of high-dimensional covariance matrices

Yilun Chen*, Ami Wiesel, Alfred O. Hero

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

164 Scopus citations

Abstract

We address high dimensional covariance estimation for elliptical distributed samples, which are also known as spherically invariant random vectors (SIRV) or compound-Gaussian processes. Specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples (large p small n). We start from a classical robust covariance estimator [Tyler (1987)], which is distribution-free within the family of elliptical distribution but inapplicable when n<p. Using a shrinkage coefficient, we regularize Tyler's fixed-point iterations. We prove that, for all n and p, the proposed fixed-point iterations converge to a unique limit regardless of the initial condition. Next, we propose a simple, closed-form and data dependent choice for the shrinkage coefficient, which is based on a minimum mean squared error framework. Simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy-tailed samples. Finally, as a real-world application we demonstrate the performance of the proposed technique in the context of activity/intrusion detection using a wireless sensor network.

Original languageAmerican English
Article number5743027
Pages (from-to)4097-4107
Number of pages11
JournalIEEE Transactions on Signal Processing
Volume59
Issue number9
DOIs
StatePublished - Sep 2011

Bibliographical note

Funding Information:
Manuscript received September 25, 2010; revised January 21, 2011; accepted March 04, 2011. Date of publication April 07, 2011; date of current version August 10, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ta-Hsin Li. This work was partially supported by AFOSR, Grant FA9550-06-1-0324. The work of A. Wiesel was supported by a Marie Curie Outgoing International Fellowship within the 7th European Community Framework Programme. Parts of this work were presented at the IEEE Workshop on Sensor Array and Multichannel Signal Processing (SAM), Jerusalem, Israel, October 2010.

Keywords

  • Activity/intrusion detection
  • covariance estimation
  • elliptical distribution
  • large p small n
  • robust estimation
  • shrinkage methods
  • wireless sensor network

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