Group symmetric robust covariance estimation

Ilya Soloveychik, Dmitry Trushin, Ami Wiesel

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

In this paper, we consider Tylers robust covariance M-estimator under group symmetry constraints. We assume that the covariance matrix is invariant to the conjugation action of a unitary matrix group, referred to as group symmetry. Examples of group symmetric structures include circulant, perHermitian, and proper quaternion matrices. We introduce a group symmetric version of Tylers estimator (STyler) and provide an iterative fixed point algorithm to compute it. The classical results claim that at least n = p + 1 sample points in general position are necessary to ensure the existence and uniqueness of Tylers estimator, where p is the ambient dimension. We show that the STyler requires significantly less samples. In some groups, even two samples are enough to guarantee its existence and uniqueness. In addition, in the case of elliptical populations, we provide high probability bounds on the error of the STyler. These, too, quantify the advantage of exploiting the symmetry structure. Finally, these theoretical results are supported by numerical simulations.

Original languageAmerican English
Article number7289442
Pages (from-to)244-257
Number of pages14
JournalIEEE Transactions on Signal Processing
Volume64
Issue number1
DOIs
StatePublished - 1 Jan 2016

Bibliographical note

Publisher Copyright:
© 2015 IEEE.

Keywords

  • Group symmetry
  • Tylers estimator
  • robust covariance matrix M-estimators
  • structured covariance estimation

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