Expected Sliced Transport Plans

TL;DR

This paper introduces Expected Sliced Transport (EST) plans, a novel method to construct explicit transportation plans from sliced optimal transport, enabling the definition of a metric between probability measures with improved computational efficiency.

Contribution

The paper proposes a new lifting operation to derive transportation plans from sliced OT, creating a metric between measures and bridging the gap between sliced OT efficiency and explicit coupling.

Findings

EST plans form a valid metric between probability measures.

The approach connects sliced OT with explicit transport plans.

Numerical examples support theoretical properties.

Abstract

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.