Random matrix-improved estimation of covariance matrix distances

Given two sets x1(1),ldots,xn1(1) and x1(2),ldots,xn2(2)∈ℝp (or mathbbCp) of random vectors with zero mean and positive definite covariance matrices C1 and C2∈ℝp× p (or mathbbCp× p), respectively, this article provides novel estimators for a wide range of distances between C1 and C2 (along with divergences between some zero mean and covariance C1 or C2 probability measures) of the form frac1psumi=1n f(λi(C1-1C2)) (with λi(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 n1,n2,p→∞ with non trivial ratios p/n1<1 and p/n2<1 (the case p/n2>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 f's of practical interest (namely, f(t)=t, f(t)=log(t), f(t)=log(1+st) and f(t)=log2(t)), a closed-form expression is provided. Beside theoretical findings, simulation results suggest an outstanding performance advantage for the proposed estimators when compared to the classical "plug-in" estimator frac1psumi=1n f(λi(hat C1-1hat C2)) (with hat Ca=frac1nasumi=1naxi(a)xi^(a)sf T), and this even for very small values of n1,n2,p.