  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.
