A subspace approach for shrinkage parameter selection in undersampled configuration for Regularised Tyler Estimators

Abstract

Regularized Tyler Estimator's (RTE) have raised attention over the past years due to their attractive performance over a wide range of noise distributions and their natural robustness to outliers. Developing adaptive methods for the selection of the regularisation parameter α is currently an active topic of research. Indeed, the bias-performance compromise of RTEs highly depends on the considered application. Thus, finding a generic rule that is optimal for every criterion and/or data configurations is not straightforward. This issue is addressed in this paper for undersampled configurations (number of samples lower than the dimension of the data). The paper proposes a new regularisation parameter selection based on a subspace reduction approach. The performance of this method is investigated in terms of estimation accuracy and for adaptive detection purposes, both on simulation and real data.

Spectral Shrinkage of Tyler's $M$-Estimator of Covariance Matrix

“…Thus, regularized M -estimators have been proposed as minimizers of a penalized M -estimation objective, which involves a regularization penalty that shrinks the estimate towards a given target matrix T [7][8][9][10]. For this class of estimators, several works were also conducted on bias-variance trade-off for regularization parameter selection (see e.g., [11] and references therein).…”