Random matrix-improved estimation of covariance matrix distances

Given two sets $x1^{{(1)},\ldots,x}{n1}^{(1)}$ and $x1^{{(2)},\ldots,x{n2}{(2)}\in\mathbb{R}p$} (or $\mathbb{C}^p$) of random vectors with zero mean and positive definite covariance matrices $C1$ and $C2\in\mathbb{R}^{p\times p}$ (or $\mathbb{C}^{p\times p}$), respectively, this article provides novel estimators for a wide range of distances between $C1$ and $C2$ (along with divergences between some zero mean and covariance $C1$ or $C2$ probability measures) of the form $\frac1p\sum{i=1}ⁿ f(\lambdai(C1^{-1}C2))$ (with $\lambdai(X)$ the eigenvalues of matrix $X$). These estimators are derived using recent advances in the field of random matrix theory and are asymptotically consistent as $n1,n2,p\to\infty$ with non trivial ratios $p/n1<1$ and $p/n_2<1$ (the case $p/n_2>1$ 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 $f(t)=\log^2(t)$), 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 $\frac1p\sum{i=1}ⁿ f(\lambdai(\hat C1^{-1}\hat C2))$ (with $\hat Ca=\frac1{na}\sum{i=1}^{na}xi^{(a)}xi^{(a){\sf T}}$), and this even for very small values of $n1,n2,p$.