Alpha Procrustes metrics between positive definite operators: a unifying formulation for the Bures-Wasserstein and Log-Euclidean/Log-Hilbert-Schmidt metrics

This work presents a parametrized family of distances, namely the Alpha Procrustes on set symmetric, positive definite (SPD) matrices. The distances provide unified formulation encompassing both Bures-Wasserstein and Log-Euclidean between SPD We show that are Riemannian corresponding to metrics manifold matrices, which encompass Wasserstein metrics. is then generalized Hilbert-Schmidt operators Hilbert space, unifying infinite-dimensional Log-Hilbert-Schmidt distances. In setting reproducing kernel spaces (RKHS) covariance operators, we obtain closed form formulas for all via Gram From statistical viewpoint, give rise Gaussian measures Euclidean in finite-dimensional case, separable spaces, 2-Wasserstein distance, with matrices RKHS setting. presented formulations new finite settings.