论文标题
在一般矩阵值得不平衡的最佳运输问题上
On a general matrix-valued unbalanced optimal transport problem
论文作者
论文摘要
我们在积极的半明确矩阵值rad子上介绍了一般的运输距离$ {\ rm wb}_λ$。这种距离是通过具有加权动作功能和抽象矩阵连续性方程的广义贝纳莫 - 布雷犬公式来定义的，这导致了凸优化问题。最近提出的一些模型，包括Kantorovich-Bures距离和Wasserstein-Fisher-Rao距离，自然可以适合我们的模型。我们给出了最小化器的完整表征，并探索了空间$（\ Mathcal {m}（ω，\ Mathbb {s} _+^n），{\ rm wb}_λ）$的拓扑和几何特性。特别是，我们表明$（\ Mathcal {m}（ω，\ Mathbb {s} _+^n），{\ rm wb}_λ）$是一个完整的地理空间，并具有圆锥结构。
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.
