New Methods for Handling Singular Sample Covariance Matrices

TL;DR

This paper introduces new methods, including the Ewens estimator, for estimating covariance matrices when data is insufficient, leveraging randomization, dimensionality reduction, and advanced mathematical techniques.

Contribution

It extends previous work by proposing novel approaches like the Ewens estimator for better handling singular covariance matrices.

Findings

Ewens estimator improves covariance estimation accuracy.

Dimensionality reduction combined with randomization enhances robustness.

The methods are grounded in random matrix theory and combinatorics.

Abstract

The estimation of a covariance matrix from an insufficient amount of data is one of the most common problems in fields as diverse as multivariate statistics, wireless communications, signal processing, biology, learning theory and finance. In a joint work of Marzetta, Tucci and Simon, a new approach to handle singular covariance matrices was suggested. The main idea was to use dimensionality reduction in conjunction with an average over the Stiefel manifold. In this paper we continue with this research and we consider some new approaches to solve this problem. One of the methods is called the Ewens estimator and uses a randomization of the sample covariance matrix over all the permutation matrices with respect to the Ewens measure. The techniques used to attack this problem are broad and run from random matrix theory to combinatorics.