Estimation of Covariance Matrix Distances in the High Dimension Low Sample Size Regime

Abstract

A broad family of distances between two covariance matrices C1, C2 ∈ R p×p , among which the Frobenhius, Fisher, Battacharrya distances as well as the Kullback-Leibler, Rényi and Wasserstein divergence for centered Gaussian distributions can be expressed as functionalsConsistent estimates of such distances based on few (n1, n2) samples xi ∈ R p having covariance C1, C2 have been recently proposed using random matrix tools in the regime where n1, n2 ∼ p. These estimates however demand that n1, n2 > p for most functions f . The article proposes to alleviate this limitation using a polynomial approximation approach. The proposed method is supported by simulations in practical applications.