Random matrix-improved estimation of covariance matrix distances
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Given two sets x_1^(1),...,x_n_1^(1) and x_1^(2),...,x_n_2^(2)∈R^p (or C^p) of random vectors with zero mean and positive definite covariance matrices C_1 and C_2∈R^p× p (or C^p× p), 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∑_i=1^n f(λ_i(C_1^-1C_2)) (with λ_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→∞ 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 f's of practical interest (namely, f(t)=t, f(t)=(t), f(t)=(1+st) and f(t)=^2(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 1/p∑_i=1^n f(λ_i(Ĉ_1^-1Ĉ_2)) (with Ĉ_a=1/n_a∑_i=1^n_ax_i^(a)x_i^(a) T), and this even for very small values of n_1,n_2,p.
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