Random matrix-improved estimation of covariance matrix distances

Abstract: Given two sets $x_1<sup>{(1)},\ldots,x_{n_1}<sup>{(1)}$ and $$x_1<sup>{(2)},\ldots,x_{n_2}<sup>{(2)}\in\mathbb{R}<sup>p$$ (or $\mathbb{C}<sup>p$) of random vectors with zero mean and positive definite covariance matrices $C_1$ and $C_2\in\mathbb{R}^{p\times p}$ (or $\mathbb{C}^{p\times p}$), respectively, this article provides novel estimators for a wide range of distances between $C_1$ and $C_2$ (along with divergences between some zero mean and covariance $C_1$ or $C_2$ probability measures) of the form $$\frac1p\sum_{i=1}ⁿ f(\lambda_i(C_1<sup>{-1}C_2))$$ (with $\lambda_i(X)$ the eigenvalues of matrix $X$). These estimators are derived using recent advances in the field of random matrix theory and are asymptotically consistent as $n_1,n_2,p\to\infty$ with non trivial ratios $p/n_1&lt;1$ and $p/n_2&lt;1$ (the case $p/n_2&gt;1$ is also discussed). A first "generic" estimator, valid for a large set of $f$ functions, is provided under the form of a complex integral. Then, for a selected set of $f$'s of practical interest (namely, $f(t)=t$, $f(t)=\log(t)$, $f(t)=\log(1+st)$ and $f(t)=\log<sup>2(t)$), a closed-form expression is provided. Beside theoretical findings, simulation results suggest an outstanding performance advantage for the proposed estimators when compared to the classical "plug-in" estimator $$\frac1p\sum_{i=1}ⁿ f(\lambda_i(\hat C_1^{-1}\hat C_2))$$ (with $$\hat C_a=\frac1{n_a}\sum_{i=1}^{{n_a}x_i^{{(a)}x_i^{(a){\sf}} T}}$$), and this even for very small values of $n_1,n_2,p$.