关于一般矩阵值非平衡最优传输问题
On a general matrix-valued unbalanced optimal transport problem
AI总结：
本文引入一类加权Wasserstein-Bures距离，统一了Kantorovich-Bures和Wasserstein-Fisher-Rao模型，并证明其空间为完备测地空间且具有锥结构。
摘要：
我们在正半定矩阵值Radon测度空间 $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ 上引入一类一般的传输距离 ${\rm WB}_\Lambda$，称为加权Wasserstein-Bures距离。该距离通过带有加权作用泛函和抽象矩阵连续性方程的广义Benamou-Brenier公式定义，从而产生一个凸优化问题。最近提出的一些模型，包括Kantorovich-Bures距离和Wasserstein-Fisher-Rao距离，可以自然地纳入我们的框架。我们给出了极小元的完整刻画，并探讨了空间 $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_\Lambda)$ 的拓扑和几何性质。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，并具有锥结构。
英文摘要：
We introduce a general class of transport distances ${\rm WB}_Λ$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(Ω,\mathbb{S}_+^n)$, 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 $(\mathcal{M}(Ω,\mathbb{S}_+^n),{\rm WB}_Λ)$. In particular, we show that $(\mathcal{M}(Ω,\mathbb{S}_+^n),{\rm WB}_Λ)$ is a complete geodesic space and exhibits a conic structure.
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