Optimal transport (OT) has become exceedingly popular in machine learning, data science, and computer vision. The core assumption in the OT problem is the equal total amount of mass in source and target measures, which limits its application. Optimal Partial Transport (OPT) is a recently proposed solution to this limitation. Similar to the OT problem, the computation of OPT relies on solving a linear programming problem (often in high dimensions), which can become computationally prohibitive. In this paper, we propose an efficient algorithm for calculating the OPT problem between two non-negative measures in one dimension. Next, following the idea of sliced OT distances, we utilize slicing to define the sliced OPT distance. Finally, we demonstrate the computational and accuracy benefits of the sliced OPT-based method in various numerical experiments. In particular, we show an application of our proposed Sliced-OPT in noisy point cloud registration.

翻译：最优输运（OT）在机器学习、数据科学和计算机视觉中已变得极为流行。OT问题的核心假设是源测度和目标测度的总质量相等，这限制了其应用。最优局部输运（OPT）是近期针对这一局限提出的解决方案。与OT问题类似，OPT的计算同样依赖于求解线性规划问题（通常在高维空间中），这可能带来极高的计算开销。本文针对一维空间中两个非负测度间的OPT问题，提出了一种高效算法。随后，借鉴切片OT距离的思想，我们利用切片方法定义了切片OPT距离。最后，通过多项数值实验展示了基于切片OPT方法的计算效率与精度优势，特别地，我们演示了所提出的Sliced-OPT在含噪点云配准中的应用。