Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered Gaussian Distributions
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This article proposes a method to consistently estimate functionals 1/p∑_i=1^pf(λ_i(C_1C_2)) of the eigenvalues of the product of two covariance matrices C_1,C_2∈R^p× p based on the empirical estimates λ_i(Ĉ_1Ĉ_2) (Ĉ_a=1/n_a∑_i=1^n_a x_i^(a)x_i^(a) T), when the size p and number n_a of the (zero mean) samples x_i^(a) are similar. As a corollary, a consistent estimate of the Wasserstein distance (related to the case f(t)=√(t)) between centered Gaussian distributions is derived. The new estimate is shown to largely outperform the classical sample covariance-based `plug-in' estimator. Based on this finding, a practical application to covariance estimation is then devised which demonstrates potentially significant performance gains with respect to state-of-the-art alternatives.
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