1A Random Matrix–Theoretic Approach to Handling Singular Covariance Estimates

Abstract

Abstract—In many practical situations we would like to es-timate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set of N independent, identically distributed measurements of an M dimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case where N < M such that this estimate is singular (non–invertible) and therefore fundamentally bad. We present a radically new approach to deal with this situation. Let X be the M ×N data matrix, where the columns are the N independent realizations of the random vector with covariance matrix Σ. Without loss of generality, and for simplicity, we can assume that the random variables have zero mean. We would like to estimate Σ from X. Let K be the classical sample covariance matrix. Fix a parameter 1 ≤ L ≤ N and consider an ensemble of L ×