Optimal transport (OT) is a popular and powerful tool for comparing probability measures. However, OT suffers a few drawbacks: (i) input measures required to have the same mass, (ii) a high computational complexity, and (iii) indefiniteness which limits its applications on kernel-dependent algorithmic approaches. To tackle issues (ii)--(iii), Le et al. (2022) recently proposed Sobolev transport for measures on a graph having the same total mass by leveraging the graph structure over supports. In this work, we consider measures that may have different total mass and are supported on a graph metric space. To alleviate the disadvantages (i)--(iii) of OT, we propose a novel and scalable approach to extend Sobolev transport for this unbalanced setting where measures may have different total mass. We show that the proposed unbalanced Sobolev transport (UST) admits a closed-form formula for fast computation, and it is also negative definite. Additionally, we derive geometric structures for the UST and establish relations between our UST and other transport distances. We further exploit the negative definiteness to design positive definite kernels and evaluate them on various simulations to illustrate their fast computation and comparable performances against other transport baselines for unbalanced measures on a graph.

翻译：最优输运（OT）是比较概率测度的一种流行且强大的工具。然而，OT存在若干缺陷：（i）要求输入测度具有相同质量，（ii）计算复杂度高，（iii）半正定性缺失限制了其在核依赖算法方法中的应用。为解决缺陷（ii）—（iii），Le等人（2022）近期提出利用支撑集上的图结构，针对具有相同总质量的图上测度提出了Sobolev输运。在本工作中，我们考虑可能具有不同总质量且支撑于图度量空间上的测度。为克服OT的缺陷（i）—（iii），我们提出一种新颖且可扩展的方法，将Sobolev输运推广至测度可能具有不同总质量的非平衡场景。我们证明所提出的非平衡Sobolev输运（UST）具有闭式公式以实现快速计算，且具有负定性。此外，我们推导了UST的几何结构，并建立了UST与其他输运距离之间的关系。我们进一步利用负定性设计正定核，并通过多种仿真实验评估其快速计算性能，证明其在图上非平衡测度场景下相较于其他输运基线方法具有相当的表现。