Published on Fri Mar 08 2019

### 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. A consistent estimate of the Wasserstein distance between centered Gaussian distributions is derived. The new estimate is shown to largely outperform the classical sample-based plug-in' estimator.

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###### Abstract

This article proposes a method to consistently estimate functionals of the eigenvalues of the product of two covariance matrices based on the empirical estimates ($\hat C_a=\frac1{n_a}\sum_{i=1}^{n_a} x_i^{(a)}x_i^{(a){{\sf T}}}$), when the size and number of the (zero mean) samples are similar. As a corollary, a consistent estimate of the Wasserstein distance (related to the case ) 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.