Infinite-Dimensional Log-Determinant Divergences III: Log-Euclidean and Log-Hilbert–Schmidt Divergences

Abstract

We introduce a family of Log-Determinant (Log-Det) divergences on the set of symmetric, positive definite (SPD) matrices that includes the Log-Euclidean distance as a special case. This is then generalized to a family of Log-Det divergences on the set of positive definite Hilbert–Schmidt operators on a Hilbert space, with the Log-Hilbert–Schmidt distance being a special case. The divergences introduced here are novel both in the finite and infinite-dimensional settings. We also generalize the Power Euclidean distances to the infinite-dimensional setting, which we call the Extended Power Hilbert–Schmidt distances. While these families include the Log-Euclidean and Log-Hilbert–Schmidt distances, respectively, as special cases, they do not satisfy the same invariances as the latter distances, in contrast to the Log-Det divergences. In the case of RKHS covariance operators, we provide closed form formulas for all of the above divergences and distances in terms of the corresponding Gram matrices.