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：最优运输(OT)在机器学习、数据科学和计算机视觉中变得越来越受欢迎。OT问题的核心假设是源和目标测量中等质量的总量，从而限制了其应用。最优偏移运输(OPT)是最近提出解决此限制的解决方案。与OT问题类似，计算OPT依赖于求解线性规划问题（通常在高维数下），这可能会变得计算上禁止。在本文中，我们提出了一种计算一维非负测量之间OPT问题的高效算法。接下来，借助切片的思想，我们利用切片来定义切片OPT距离。最后，我们在各种数值实验中展示了切片OPT方法的计算和准确性优势。特别是，我们展示了我们提出的切片-OPT在噪声点云配准中的应用。