Random matrix-improved estimation of covariance matrix distances

Given two sets x1(1),…,xn1(1) and x1(2),…,xn2(2)∈Rp (or ℂp) of random vectors with zero mean and positive definite covariance matrices C1 and C2∈Rp×p (or ℂp×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 1p∑i=1nf(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 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)=ln2(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 1p∑i=1nf(λi(Cˆ1−1Cˆ2)) (with Cˆa=1na∑i=1naxi(a)xi(a)⊤), and this even for very small values of n1,n2,p. A concrete application to kernel spectral clustering of covariance classes supports this claim.