Tree-Sliced Wasserstein Distance: A Geometric Perspective

TL;DR

This paper introduces Tree-Sliced Wasserstein distance, a new geometric approach that improves upon traditional sliced Wasserstein by using tree structures to better preserve topological information, with applications in image processing and generative models.

Contribution

It proposes a novel tree-based structure for optimal transport, providing a closed-form metric and theoretical analysis, enhancing the effectiveness of Wasserstein computations.

Findings

Outperforms sliced Wasserstein in experiments

Preserves topological information better

Efficient computation of measures on tree systems

Abstract

Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.