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

• Published in: Linear algebra and its applications 2022-03, Vol.636, p.25-68
• Main Author: Minh, Hà Quang
• Format: Article
• Language: English
• Subjects: Closed form solutions; Covariance operators; Euclidean geometry; Euclidean space; Exact solutions; Formulations; Gaussian measures; Hilbert space; Kernels; Linear algebra; Log-Euclidean distance; Log-Hilbert-Schmidt distance; Mathematical analysis; Matrices (mathematics); Operators (mathematics); Parameterization; Positive definite matrices and operators; Procrustes distance; Reproducing kernel Hilbert spaces; Wasserstein distance
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2021.11.011

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.