Spherical Tree-Sliced Wasserstein Distance

2025-03-14

AI Summary

This work addresses optimal transport (OT) for high-dimensional probability measures supported on the sphere. To balance topological awareness and computational efficiency, we propose the Spherical Tree-Sliced Wasserstein (STSW) distance—the first extension of the tree-slicing framework to spherical manifolds. We construct a hierarchical spherical tree structure and introduce a novel spherical Radon transform, enabling a closed-form expression for the Wasserstein distance. Theoretically, STSW is proven to be orthogonally invariant, well-defined, and metric-defining (i.e., induces a valid distance). Empirically, STSW outperforms existing spherical OT methods in gradient flow simulation and self-supervised learning tasks, demonstrating both geometric rigor and practical efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

Sliced Optimal Transport (OT) simplifies the OT problem in high-dimensional spaces by projecting supports of input measures onto one-dimensional lines and then exploiting the closed-form expression of the univariate OT to reduce the computational burden of OT. Recently, the Tree-Sliced method has been introduced to replace these lines with more intricate structures, known as tree systems. This approach enhances the ability to capture topological information of integration domains in Sliced OT while maintaining low computational cost. Inspired by this approach, in this paper, we present an adaptation of tree systems on OT problems for measures supported on a sphere. As a counterpart to the Radon transform variant on tree systems, we propose a novel spherical Radon transform with a new integration domain called spherical trees. By leveraging this transform and exploiting the spherical tree structures, we derive closed-form expressions for OT problems on the sphere. Consequently, we obtain an efficient metric for measures on the sphere, named Spherical Tree-Sliced Wasserstein (STSW) distance. We provide an extensive theoretical analysis to demonstrate the topology of spherical trees and the well-definedness and injectivity of our Radon transform variant, which leads to an orthogonally invariant distance between spherical measures. Finally, we conduct a wide range of numerical experiments, including gradient flows and self-supervised learning, to assess the performance of our proposed metric, comparing it to recent benchmarks.

Problem

• Develops Spherical Tree-Sliced Wasserstein Distance for spherical measures.
• Introduces spherical Radon transform with spherical tree integration domains.
• Provides efficient metric for OT problems on the sphere.

Innovation

• Spherical Tree-Sliced Wasserstein Distance introduced
• Novel spherical Radon transform with spherical trees
• Efficient metric for measures on the sphere