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：文献[arXiv:2011.05845]以Benamou-Brenier方式定义了矩阵值测度$\mathcal{M}(\Omega,\mathbb{S}_+^n)$上的广义非平衡最优传输距离${\rm WB}_{\Lambda}$，该距离推广了Kantorovich-Bures距离和Wasserstein-Fisher-Rao距离。本文研究了与${\rm WB}_{\Lambda}$相关的离散传输问题的收敛性质。我们首先提出了一个适用于抽象离散化的收敛框架。随后，在初始和终末分布关于Lebesgue测度绝对连续的假设下，我们提出了一种与该框架相兼容的具体离散化方案。此外，借助静态形式化方法，我们证明对于Wasserstein-Fisher-Rao距离可以去除上述假设条件。