COVARIANCE 2017

The set of SPD matrices is a not vector subspace of Euclidean space under the standard matrix addition and scalar multiplication operations, but is an open convex cone that also admits a smooth manifold structure. Consequently, in general, the optimal measure of similarity between covariance matrices is not the Euclidean distance, but a distance that captures the intrinsic geometry of SPD matrices. Among the most widely used non-Euclidean distances for SPD matrices are the classical affine-invariant Riemannian distance, the recently introduced Log-Euclidean distance, and the distance-like Bregman divergences. All of these distances and divergences have recently been generalized to the infinite-dimensional setting. In the case of RKHS covariance operators, they admit closed form formulas via the corresponding Gram matrices. For large scale data sets, which require the handling of large numbers of covariance operators, these distances and divergences can be approximated by kernel..