Highly Data Parallelizable Estimation of the Sliced-Wasserstein Distance Using Cumulative Distribution Functions

2026-06-29

AI Summary

This work addresses the limitations of the traditional Sliced-Wasserstein distance, which relies on sample sorting and access to the full dataset, thereby hindering parallelization and compromising privacy. The authors introduce, for the first time, a novel estimator that leverages cumulative distribution functions (CDFs) to eliminate the need for explicit sorting. By combining random projections with one-dimensional optimal transport theory, the proposed method enables large-scale parallel computation and local aggregation. It is inherently compatible with distributed frameworks such as federated learning and achieves substantial gains in computational efficiency for distributions with tractable CDFs—e.g., Gaussian mixtures—while simultaneously enhancing scalability and privacy preservation.

Abstract

The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.

Problem

• Sliced Wasserstein distance
• cumulative distribution function
• data parallelism
• federated learning
• quantile function

Innovation

• Sliced-Wasserstein distance
• cumulative distribution function
• data parallelism
• federated learning
• optimal transport