Sliced-Regularized Optimal Transport

TL;DR

This paper introduces Sliced-Regularized Optimal Transport (SROT), a novel regularization method that improves OT plan approximation by leveraging a smoothened sliced OT plan, with efficient algorithms and superior performance demonstrated in experiments.

Contribution

The paper presents the first formulation of SROT, combining sliced OT with regularization, along with a scalable Sinkhorn-style algorithm and a new divergence measure, outperforming existing methods.

Findings

SROT yields more accurate OT plans than entropic OT under the same regularization.

The SROT divergence has favorable topological and computational properties.

Experiments show SROT outperforms EOT and SOT in synthetic and color transfer tasks.

Abstract

We propose a new regularized optimal transport (OT) formulation, termed sliced-regularized optimal transport (SROT). Unlike entropic OT (EOT), which regularizes the transport plan toward an independent coupling, SROT regularizes it toward a smoothened sliced OT (SOT) plan. To the best of our knowledge, SROT is the first approach to leverage a version of SOT plan as a reference to improve classical OT. We provide a formal definition of SROT, derive its dual formulation, and provide a post-Bayesian interpretation of SROT. We then develop a Sinkhorn-style algorithm for efficient computation, retaining the same scalability advantages as EOT. By incorporating a scalable SOT plan as a prior, SROT yields more accurate approximations of the exact OT plan than EOT under the same level of regularization. Moreover, the resulting transport plan improves upon the reference SOT plan itself. We further introduce the corresponding OT divergence induced by SROT, named SROT divergence, and analyze its topological and computational properties. Finally, we validate our approach through experiments on synthetic datasets and color transfer tasks, demonstrating that SROT is better than both EOT and SOT in approximating exact OT. Additional experiments on gradient flows further highlight the advantages of SROT divergence.