Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry
This work addresses a domain-specific problem in machine learning for applications involving symmetric positive definite matrices, and it is incremental as it builds on the existing Bures-Wasserstein geometry.
The authors tackled the problem of learning with symmetric positive definite matrices by proposing a novel generalization of the Bures-Wasserstein geometry, called GBW geometry, which is parameterized by a matrix and recovers the original when set to identity, and they demonstrated its efficacy over the existing geometry in experiments.
Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix $\mathbf{M}$ such that when $\mathbf{M} = \mathbf{I}$, we recover the BW geometry. We provide a rigorous treatment to study various differential geometric notions on the proposed novel generalized geometry which makes it amenable to various machine learning applications. We also present experiments that illustrate the efficacy of the proposed GBW geometry over the BW geometry.