The optimal transport (OT) problem has gained significant traction in modern machine learning for its ability to: (1) provide versatile metrics, such as Wasserstein distances and their variants, and (2) determine optimal couplings between probability measures. To reduce the computational complexity of OT solvers, methods like entropic regularization and sliced optimal transport have been proposed. The sliced OT framework improves efficiency by comparing one-dimensional projections (slices) of high-dimensional distributions. However, despite their computational efficiency, sliced-Wasserstein approaches lack a transportation plan between the input measures, limiting their use in scenarios requiring explicit coupling. In this paper, we address two key questions: Can a transportation plan be constructed between two probability measures using the sliced transport framework? If so, can this plan be used to define a metric between the measures? We propose a "lifting" operation to extend one-dimensional optimal transport plans back to the original space of the measures. By computing the expectation of these lifted plans, we derive a new transportation plan, termed expected sliced transport (EST) plans. We prove that using the EST plan to weight the sum of the individual Euclidean costs for moving from one point to another results in a valid metric between the input discrete probability measures. We demonstrate the connection between our approach and the recently proposed min-SWGG, along with illustrative numerical examples that support our theoretical findings.

本研究解决了切片最优传输框架在输入概率测度之间缺乏运输计划的问题，提出了一种“提升”操作，将一维最优传输计划扩展回原测度空间。通过计算这些提升计划的期望，提出了期望切片传输（EST）计划，证明了该计划可用于在输入离散概率测度之间定义有效度量。

期望切片传输计划