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.

翻译：对称正定（SPD）矩阵学习在机器学习中具有广泛应用。因此，理解SPD矩阵的黎曼几何近来备受关注。其中一种感兴趣的黎曼几何是最近提出的Bures-Wasserstein（BW）几何，它建立在高斯密度之间的Wasserstein距离基础上。本文提出了BW几何的一种新推广，称为GBW几何。该推广由一个对称正定矩阵$\mathbf{M}$参数化，当$\mathbf{M} = \mathbf{I}$时，我们恢复为BW几何。我们对该新广义几何上的多种微分几何概念进行了严格处理，使其适用于各类机器学习应用。我们还通过实验展示了所提出的GBW几何相对于BW几何的有效性。