Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered Gaussian Distributions

This article proposes a method to consistently estimate functionals frac1psumi=1pf(λi(C1C2)) of the eigenvalues of the product of two covariance matrices C1,C2∈ℝp× p based on the empirical estimates λi(hat C1hat C2) (hat Ca=frac1nasumi=1na xi(a)xi^(a)sf T), when the size p and number na of the (zero mean) samples xi(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.