The random matrix regime of Maronna's M-estimator for observations corrupted by elliptical noises

TL;DR

This paper analyzes the behavior of Maronna's M-estimator in high-dimensional settings with heavy-tailed elliptical noise, showing it converges to a standard random matrix model under certain growth conditions.

Contribution

It provides a theoretical characterization of Maronna's M-estimator in the large-dimensional regime with elliptical noise, extending understanding of its asymptotic behavior.

Findings

Robust scatter estimator converges to a standard random matrix model.

Analysis applicable to high-dimensional statistical inference and signal processing.

Provides asymptotic characterization under mild assumptions.

Abstract

This article studies the behavior of the Maronna robust scatter estimator $\hat{C}_N\in \mathbb{C}^{N\times N}$ of a sequence of observations $y_1,...,y_n$ which is composed of a $K$ dimensional signal drown in a heavy tailed noise, i.e $y_i=A_N s_i+x_i$ where $A_N \in \mathbb{C}^{N\times K}$ and $x_i$ is drawn from elliptical distribution. In particular, we prove that as the population dimension $N$, the number of observations $n$ and the rank of $A_N$ grow to infinity at the same pace and under some mild assumptions, the robust scatter matrix can be characterized by a random matrix $\hat{S}_N$ that follows a standard random model. Our analysis can be very useful for many applications of the fields of statistical inference and signal processing.