A generalized unbalanced optimal transport distance ${\rm WB}_{\Lambda}$ on matrix-valued measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ was defined in [arXiv:2011.05845] \`{a} la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with ${\rm WB}_{\Lambda}$. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, under the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Moreover, thanks to the static formulation, we show that such an assumption can be removed for the Wasserstein-Fisher-Rao distance.

翻译：在[arXiv:2011.05845]中，以Benamou-Brenier方式在矩阵值测度空间$\mathcal{M}(\Omega,\mathbb{S}_+^n)$上定义了一种广义非平衡最优输运距离${\rm WB}_{\Lambda}$，该距离推广了Kantorovich-Bures距离与Wasserstein-Fisher-Rao距离。本文研究了与${\rm WB}_{\Lambda}$相关的离散输运问题的收敛性质。我们首先提出了抽象离散化的一般收敛框架，随后在假设初始分布与终值分布均关于Lebesgue测度绝对连续的条件下，给出了一种符合该框架的具体离散化方案。此外，借助静态表述形式，我们证明了对于Wasserstein-Fisher-Rao距离，该绝对连续性假设可以被移除。