Robust sparse covariance estimation by thresholding Tyler's m-estimator

John Goes, Gilad Lerman, Boaz Nadler

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental task in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Toward bridging this gap, in this work we consider estimating a sparse shape matrix from n samples following a possibly heavy-tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We prove that in the joint limit as the dimension p and the sample size n tend to infinity with p/n → γ > 0, our estimators are minimax rate optimal. Results on simulated data support our theoretical analysis.

Original languageEnglish (US)
Pages (from-to)86-110
Number of pages25
JournalAnnals of Statistics
Volume48
Issue number1
DOIs
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© Institute of Mathematical Statistics, 2020

Keywords

  • Elliptical distribution
  • Matrix estimation
  • Sparsity
  • Spectral norm
  • Thresholding
  • Tyler's M-estimator

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