We 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.

翻译：我们引入了一类定义在半正定矩阵值 Radon 测度空间 $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ 上的输运距离 ${\rm WB}_{\Lambda}$，称为加权 Wasserstein-Bures 距离。该距离通过广义的 Benamou-Brenier 公式定义，包含一个加权作用量泛函和一个抽象的矩阵连续性方程，从而导出一个凸优化问题。一些近期的模型，包括 Kantorovich-Bures 距离和 Wasserstein-Fisher-Rao 距离，可以自然地归入我们的框架。我们给出了极小化子的完整刻画，并探讨了空间 $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ 的拓扑与几何性质。特别地，我们证明了 $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ 是一个完备的测地空间，并展现出圆锥结构。