This paper studies the Partial Optimal Transport (POT) problem between two unbalanced measures with at most $n$ supports and its applications in various AI tasks such as color transfer or domain adaptation. There is hence the need for fast approximations of POT with increasingly large problem sizes in arising applications. We first theoretically and experimentally investigate the infeasibility of the state-of-the-art Sinkhorn algorithm for POT due to its incompatible rounding procedure, which consequently degrades its qualitative performance in real world applications like point-cloud registration. To this end, we propose a novel rounding algorithm for POT, and then provide a feasible Sinkhorn procedure with a revised computation complexity of $\mathcal{\widetilde O}(n^2/\varepsilon^4)$. Our rounding algorithm also permits the development of two first-order methods to approximate the POT problem. The first algorithm, Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD), finds an $\varepsilon$-approximate solution to the POT problem in $\mathcal{\widetilde O}(n^{2.5}/\varepsilon)$, which is better in $\varepsilon$ than revised Sinkhorn. The second method, Dual Extrapolation, achieves the computation complexity of $\mathcal{\widetilde O}(n^2/\varepsilon)$, thereby being the best in the literature. We further demonstrate the flexibility of POT compared to standard OT as well as the practicality of our algorithms on real applications where two marginal distributions are unbalanced.

翻译：本文研究了两个最多具有$n$个支撑点的不平衡测度之间的部分最优输运（POT）问题及其在各类人工智能任务（如颜色迁移或域自适应）中的应用。因此，随着应用场景中问题规模的日益增大，亟需快速近似求解POT的方法。我们首先从理论和实验两方面研究了当前最先进的Sinkhorn算法在POT中的不可行性，其根源在于不兼容的舍入步骤，进而导致该算法在实际应用（如点云配准）中性能退化。为此，我们提出了一种新型POT舍入算法，并在此基础上给出了可行的Sinkhorn过程，其修正计算复杂度为$\mathcal{\widetilde O}(n^2/\varepsilon^4)$。我们的舍入算法还支持开发两种一阶方法来近似求解POT问题。第一种算法——自适应原始-对偶加速梯度下降（APDAGD），能在$\mathcal{\widetilde O}(n^{2.5}/\varepsilon)$内找到POT问题的$\varepsilon$-近似解，在$\varepsilon$依赖关系上优于修正后的Sinkhorn。第二种方法——对偶外推法，实现了$\mathcal{\widetilde O}(n^2/\varepsilon)$的计算复杂度，成为文献中最佳算法。我们进一步通过两个边际分布不平衡的实际应用，展示了POT相较于标准OT的灵活性以及我们算法的实用性。