# Robust Sparse Covariance Estimation by Thresholding Tyler's M-Estimator

## TL;DR

This paper introduces robust methods for estimating high-dimensional sparse covariance matrices using Tyler's M-estimator, effective even with heavy-tailed data, and provides theoretical guarantees on their accuracy.

## Contribution

It proposes thresholded Tyler's M-estimators for robust sparse covariance estimation with proven minimax optimal spectral norm bounds.

## Key findings

- Bounds on spectral norm difference are minimax rate-optimal.
- Simulated data support theoretical results.
- Methods are robust to heavy-tailed distributions.

## Abstract

Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards 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 derive bounds on the difference in spectral norm between our estimators and the shape matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\to\gamma>0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.