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：本文研究两个支撑数至多为$n$的非平衡测度之间的部分最优运输（Partial Optimal Transport, POT）问题及其在颜色迁移、域适应等人工智能任务中的应用。随着实际问题规模的不断增大，快速近似POT方法的需求日益迫切。我们首先从理论与实验两方面论证了现有最先进的Sinkhorn算法因舍入过程不兼容而无法求解POT问题，进而导致其在点云配准等实际应用中性能退化。为此，我们提出了一种新的POT舍入算法，并基于修正后的计算复杂度$\mathcal{\widetilde O}(n^2/\varepsilon^4)$给出了可行的Sinkhorn求解流程。该舍入算法还支持两种一阶方法近似POT问题：第一种方法——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$\varepsilon$近似解，其$\varepsilon$阶优于修正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，为当前文献中最优结果。通过两个边缘分布非平衡的实际应用，我们进一步证明了POT相较于标准OT的灵活性以及所提算法的实用性。