Differentiable Generalized Sliced Wasserstein Plans

TL;DR

This paper introduces a differentiable reformulation of the min-SWGG slicing scheme for optimal transport, enabling efficient high-dimensional and manifold data applications, including image generation.

Contribution

It proposes a bilevel optimization and differentiable approximation for the min-SWGG scheme, extending it to manifolds and improving computational efficiency in high dimensions.

Findings

Effective in high-dimensional spaces

Applicable to data on manifolds

Enhances image generation methods

Abstract

Optimal Transport (OT) has attracted significant interest in the machine learning community, not only for its ability to define meaningful distances between probability distributions -- such as the Wasserstein distance -- but also for its formulation of OT plans. Its computational complexity remains a bottleneck, though, and slicing techniques have been developed to scale OT to large datasets. Recently, a novel slicing scheme, dubbed min-SWGG, lifts a single one-dimensional plan back to the original multidimensional space, finally selecting the slice that yields the lowest Wasserstein distance as an approximation of the full OT plan. Despite its computational and theoretical advantages, min-SWGG inherits typical limitations of slicing methods: (i) the number of required slices grows exponentially with the data dimension, and (ii) it is constrained to linear projections. Here, we reformulate min-SWGG as a bilevel optimization problem and propose a differentiable approximation scheme to efficiently identify the optimal slice, even in high-dimensional settings. We furthermore define its generalized extension for accommodating to data living on manifolds. Finally, we demonstrate the practical value of our approach in various applications, including gradient flows on manifolds and high-dimensional spaces, as well as a novel sliced OT-based conditional flow matching for image generation -- where fast computation of transport plans is essential.