This paper proposes a generalized Bures-Wasserstein (BW) Riemannian geometry for the manifold of symmetric positive definite matrices. We explore the generalization of the BW geometry in three different ways: 1) by generalizing the Lyapunov operator in the metric, 2) by generalizing the orthogonal Procrustes distance, and 3) by generalizing the Wasserstein distance between the Gaussians. We show that they all lead to the same 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 derive expressions for the distance, geodesic, exponential/logarithm maps, Levi-Civita connection, and sectional curvature under the generalized BW geometry. We also present applications and experiments that illustrate the efficacy of the proposed geometry.

翻译：本文建议对正对正确定矩阵的方方面面采用通用的布雷斯-沃塞尔斯坦(BW)里曼尼学几何测量法。我们探讨BW几何学的概括化有三种不同方式:(1) 通过在衡量标准中将Lyapunov操作员加以概括化,(2) 通过对正正对正对正对矩阵的距离进行概括化,(3) 通过对高斯人之间的瓦塞斯坦距离进行概括化, 我们显示它们都会导致相同的几何学。 拟议的一般化由正对正对正矩阵($\mathbf{M}$)进行参数化参数化。 当 $\mathbf{M} =\mathbf{I}$时, 我们恢复BW几何学。 我们从一般的BW几何学中得出距离、 地标、 指数/ logism 地图、 levi- Civita 连接和 区域曲线的表达方式。我们还介绍了用来说明拟议几何效果的应用和实验。