Distance-Based Tree-Sliced Wasserstein Distance

2025-03-14

AI Summary

Existing Tree-Sliced Wasserstein on Systems of Lines (TSW-SL) methods rely solely on support point locations while ignoring the projection domain, and their fixed partitioning mappings lack Euclidean invariance, hindering preservation of high-dimensional topological structure. Method: We propose Distance-aware tree-sliced Wasserstein (Db-TSW), which introduces a class of distance-sensitive generalized splitting mappings that explicitly encode full positional information of the input metric, ensuring Euclidean invariance. We establish a Radon-transform-based theoretical framework, proving Db-TSW’s injectivity and metric validity, and design GPU-efficient tree sampling and distance computation mechanisms. Contribution/Results: Experiments demonstrate that Db-TSW significantly outperforms mainstream Sliced Wasserstein variants across diverse tasks—achieving notable accuracy gains—while maintaining linear time complexity and low computational overhead.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

To overcome computational challenges of Optimal Transport (OT), several variants of Sliced Wasserstein (SW) has been developed in the literature. These approaches exploit the closed-form expression of the univariate OT by projecting measures onto (one-dimensional) lines. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL) has emerged as a promising alternative that replaces these lines with a more advanced structure called tree systems. The tree structures enhance the ability to capture topological information of the metric while preserving computational efficiency. However, at the core of TSW-SL, the splitting maps, which serve as the mechanism for pushing forward measures onto tree systems, focus solely on the position of the measure supports while disregarding the projecting domains. Moreover, the specific splitting map used in TSW-SL leads to a metric that is not invariant under Euclidean transformations, a typically expected property for OT on Euclidean space. In this work, we propose a novel class of splitting maps that generalizes the existing one studied in TSW-SL enabling the use of all positional information from input measures, resulting in a novel Distance-based Tree-Sliced Wasserstein (Db-TSW) distance. In addition, we introduce a simple tree sampling process better suited for Db-TSW, leading to an efficient GPU-friendly implementation for tree systems, similar to the original SW. We also provide a comprehensive theoretical analysis of proposed class of splitting maps to verify the injectivity of the corresponding Radon Transform, and demonstrate that Db-TSW is an Euclidean invariant metric. We empirically show that Db-TSW significantly improves accuracy compared to recent SW variants while maintaining low computational cost via a wide range of experiments.

Problem

• Overcomes computational challenges in Optimal Transport.
• Enhances topological information capture using tree systems.
• Ensures Euclidean invariance and improves accuracy efficiently.

Innovation

• Introduces Distance-based Tree-Sliced Wasserstein (Db-TSW) distance.
• Develops novel splitting maps using all positional information.
• Provides efficient GPU-friendly implementation for tree systems.