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