Robust regularized covariance matrix estimation: well-posedness and convergent algorithm

TL;DR

This paper investigates the properties of penalized multivariate scatter estimators, establishes conditions for their existence and uniqueness, and introduces a new convergent re-weighting algorithm.

Contribution

It provides theoretical conditions for estimator well-posedness and proposes a novel algorithm with guaranteed convergence for penalized scatter estimation.

Findings

Existence and uniqueness conditions are established.

Standard fixed-point algorithms may not converge for penalized estimators.

A new re-weighting algorithm with proven monotone convergence is developed.

Abstract

In this paper, we study properties of penalized and structured M-estimators of multivariate scatter, based on geodesically convex but not necessarily smooth penalty functions. Existence and uniqueness conditions for these penalized and structured estimators are given. However, we show that the standard fixed-point algorithm which is usually applied to an M-estimation problem does not necessarily converge for penalized M-estimation problems. Hence, we develop a new but simple re-weighting algorithm and prove that it has monotone convergence for a broad class of penalized and structured M-estimators of multivariate scatter.