Regularized Tyler's Scatter Estimator: Existence, Uniqueness, and Algorithms

TL;DR

This paper studies the properties and algorithms of regularized Tyler's scatter estimators for elliptical distributions, providing conditions for their existence, uniqueness, and convergence, and demonstrating their effectiveness in small sample scenarios.

Contribution

It offers a comprehensive analysis of the existence, uniqueness, and convergence of shrinkage Tyler's estimators, including new necessary conditions and algorithmic frameworks.

Findings

Regularization reduces sample size requirements for estimation.

Conditions for estimator existence and uniqueness are established.

Algorithms based on majorization-minimization converge systematically.

Abstract

This paper considers the regularized Tyler's scatter estimator for elliptical distributions, which has received considerable attention recently. Various types of shrinkage Tyler's estimators have been proposed in the literature and proved work effectively in the "small n large p" scenario. Nevertheless, the existence and uniqueness properties of the estimators are not thoroughly studied, and in certain cases the algorithms may fail to converge. In this work, we provide a general result that analyzes the sufficient condition for the existence of a family of shrinkage Tyler's estimators, which quantitatively shows that regularization indeed reduces the number of required samples for estimation and the convergence of the algorithms for the estimators. For two specific shrinkage Tyler's estimators, we also proved that the condition is necessary and the estimator is unique. Finally, we show that the two estimators are actually equivalent. Numerical algorithms are also derived based on the majorization-minimization framework, under which the convergence is analyzed systematically.