We first introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a distance is defined via a generalized Benamou-Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimization problem. Some recently proposed models, including the Kantorovich-Bures distance and the Wasserstein-Fisher-Rao distance, can naturally fit into ours. We give a complete characterization of the minimizer and explore the topological and geometrical properties of the space $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$. In particular, we show that $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ is a complete geodesic space and exhibits a conic structure. We further investigate the convergence property of the associated discrete transport problem. We present a convergence framework for abstract discretization and then propose a concrete convergent discretization scheme.

翻译：本文首先在正半定矩阵值Radon测度空间$\mathcal{M}(\Omega,\mathbb{S}_+^n)$上引入一类广义传输距离${\rm WB}_{\Lambda}$，称为加权Wasserstein-Bures距离。该距离通过广义Benamou-Brenier公式定义，包含加权作用泛函和抽象矩阵连续性方程，从而导出凸优化问题。近期提出的若干模型（包括Kantorovich-Bures距离和Wasserstein-Fisher-Rao距离）均可自然纳入该框架。我们完整刻画了极小解的性质，并深入探讨空间$(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$的拓扑与几何结构。特别地，证明$(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$是一个完备测地线空间且具有锥形结构。进一步研究相关离散传输问题的收敛性，提出抽象离散化收敛框架，并给出具体收敛离散化方案。