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：最优传输（OT）已在机器学习、数据科学和计算机视觉领域日益普及。OT问题的核心假设是源测度与目标测度中总质量相等，这限制了其应用范围。最优部分传输（OPT）是近期针对该限制提出的解决方案。与OT问题类似，OPT的计算依赖于求解线性规划问题（通常在高维空间中），这可能导致计算代价过高。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。继而借鉴切片OT距离的思想，我们利用切片技术定义了切片OPT距离。最后，通过多项数值实验验证了基于切片OPT方法的计算效率与精度优势，并特别展示了所提出的切片OPT在含噪点云配准中的应用。