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

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Malik Tiomoko
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Abstract

This article proposes a method to consistently estimate functionals 1 p p i=1 f (λi(C1C2)) of the eigenvalues of the product of two covariance matrices C1, C2 ∈ R p×p based on the empirical estimates λi(Ĉ1Ĉ2) (Ĉa = 1 na na i=1 x (a) i x (a)T i), when the size p and number na of the (zero mean) samples x (a) i 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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Dates and versions

hal-02965778 , version 1 (19-10-2020)

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Malik Tiomoko, Romain Couillet. Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered Gaussian Distributions. EUSIPCO 2019 - 27th European Signal Processing Conference, Sep 2019, A Coruna, Spain. pp.1-5, ⟨10.23919/EUSIPCO.2019.8902795⟩. ⟨hal-02965778⟩
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