Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random ProjectionsDownload PDF

Published: 09 Nov 2021, Last Modified: 16 Mar 2025NeurIPS 2021 PosterReaders: Everyone
Keywords: optimal transport, central limit theorem, random projections, high-dimensional data, generative modeling
Abstract: The Sliced-Wasserstein distance (SW) is being increasingly used in machine learning applications as an alternative to the Wasserstein distance and offers significant computational and statistical benefits. Since it is defined as an expectation over random projections, SW is commonly approximated by Monte Carlo. We adopt a new perspective to approximate SW by making use of the concentration of measure phenomenon: under mild assumptions, one-dimensional projections of a high-dimensional random vector are approximately Gaussian. Based on this observation, we develop a simple deterministic approximation for SW. Our method does not require sampling a number of random projections, and is therefore both accurate and easy to use compared to the usual Monte Carlo approximation. We derive nonasymptotical guarantees for our approach, and show that the approximation error goes to zero as the dimension increases, under a weak dependence condition on the data distribution. We validate our theoretical findings on synthetic datasets, and illustrate the proposed approximation on a generative modeling problem.
Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.
TL;DR: We develop a novel method to efficiently compute the Sliced-Wasserstein distance using concentration properties of random projections instead of the traditional Monte Carlo estimation, and we illustrate its advantages in theory and practice.
Supplementary Material: zip
Code: https://github.com/kimiandj/fast_sw
Community Implementations: [![CatalyzeX](/images/catalyzex_icon.svg) 1 code implementation](https://www.catalyzex.com/paper/fast-approximation-of-the-sliced-wasserstein/code)
15 Replies

Loading

OpenReview is a long-term project to advance science through improved peer review with legal nonprofit status. We gratefully acknowledge the support of the OpenReview Sponsors. © 2025 OpenReview