Random matrix-improved estimation of covariance matrix distances

Given two sets x(1)((1)), ..., x(n1)((1) )and x(1)((2)()), ..., x(n2)((2)) is an element of R-P (or C-p) of random vectors with zero mean and positive definite covariance matrices C-1 and C-2 is an element of R-PxP (or C-p(xp)), respectively, this article provides novel estimators for a wide range of distances between C-1 and C-2 (along with divergences between some zero mean and covariance C-1 or C-2 probability measures) of the form 1/p Sigma(n)(i=1) f(lambda(i)(C1-1C2)) (with lambda(i) (X) the eigenvalues of matrix X). These estimators are derived using recent advances in the field of random matrix theory and are asymptotically consistent as n(1), n(2), p -> infinity with non trivial ratios p/n(1) < 1 and p/n(2) < 1 (the case p/n(2) > 1 is also discussed). A first "generic" estimator, valid for a large set of f functions, is provided under the form of a complex integral. Then, for a selected set of atomic functions f which can be linearly combined into elaborate distances of practical interest (namely, f(t) = t, f(t) = ln(t), f (t) = ln(1 + st) and f(t) = ln(2)(t)), a closed-form expression is provided. Besides theoretical findings, simulation results suggest an outstanding performance advantage for the proposed estimators when compared to the classical "plug-in" estimator 1/p Sigma(n)(i=1) f(lambda(i)((C) over cap (-1)(1)(C) over cap (2))) (with (C) over cap (a) = 1/n(a) Sigma(na )(i=1)x(i)((a))x(i)((a)T)), and this even for very small values of n(1), n(2), p. A concrete application to kernel spectral clustering of covariance classes supports this claim. (C) 2019 Elsevier Inc. All rights reserved.