ESTIMATION OF COVARIANCE MATRIX DISTANCES IN THE HIGH DIMENSION LOW SAMPLE SIZE REGIME

A broad family of distances between two covariance matrices C-1, C-2 is an element of R-PxP, among which the Frobenhius, Fisher, Battacharrya distances as well as the Kullback-Leibler, Renyi and Wasserstein divergence for centered Gaussian distributions can be expressed as functionals 1/p Sigma(p)(i=1) f (lambda(i) (C-1(-1) C-2)) or 1/p Sigma(p)(i=1) f (lambda(i) (C1C2)) of the eigenvalue distribution of C-1(-1) C-2 or C-1 C-2. Consistent estimates of such distances based on few (n(1), n(2)) samples x(i) is an element of R-P having covariance C-1, C-2 have been recently proposed using random matrix tools in the regime where n(1), n(2) similar to p. These estimates however demand that n(1), n(2) > p for most functions f. The article proposes to alleviate this limitation using a polynomial approximation approach. The proposed method is supported by simulations in practical applications.