Large Dimensional Analysis of Robust M-Estimators of Covariance with Outliers

TL;DR

This paper provides a large-dimensional analysis of robust M-estimators of covariance in the presence of outliers, revealing their asymptotic behavior and effectiveness in outlier attenuation based on outlier alignment.

Contribution

It offers a novel large-dimensional asymptotic characterization of robust M-estimators with outliers, highlighting their structure and outlier rejection capabilities.

Findings

M-estimators behave like weighted sums of outer products asymptotically.

Outlier impact depends on alignment with inverse covariance matrix.

Huber estimator is most effective at rejecting outliers.

Abstract

A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, referred to as outliers. Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymptotically behaves similar to well-known random matrices when the population and sample sizes grow together to infinity. The introduction of outliers leads the robust estimator to behave asymptotically as the weighted sum of the sample outer products, with a constant weight for all legitimate samples and different weights for the outliers. A fine analysis of this structure reveals importantly that the propensity of the M-estimator to attenuate (or enhance) the impact of outliers is mostly dictated by the alignment of the outliers with the inverse population covariance matrix of the legitimate samples. Thus, robust M-estimators can bring substantial benefits over more simplistic estimators such as the per-sample normalized version of the sample covariance matrix, which is not capable of differentiating the outlying samples. The analysis shows that, within the class of Maronna's estimators of scatter, the Huber estimator is most favorable for rejecting outliers. On the contrary, estimators more similar to Tyler's scale invariant estimator (often preferred in the literature) run the risk of inadvertently enhancing some outliers.