Papers › Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered...
Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered Gaussian Distributions
Malik Tiomoko, Romain Couillet
This article proposes a method to consistently estimate functionals 1/p∑ᵢ₌₁ᵖf(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∑ᵢ₌₁^(nₐ) xᵢ⁽ᵃ⁾xᵢ⁽ᵃ⁾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.
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