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

This article proposes a method to consistently estimate functionals 1/pΣ
_i=1
^p
f(λ
_i
(C
₁
C
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
₁
Ĉ
₂
)(Ĉ
_a
= 1/n
_a
Σ
_i=1
^na
x
_i
^(a)
x
_i
^(a)
), 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.