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Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections

Kimia Nadjahi 1, 2, 3 Alain Durmus 4 Pierre E. Jacob 5 Roland Badeau 1, 2, 3 Umut Şimşekli 6 
2 S2A - Signal, Statistique et Apprentissage
LTCI - Laboratoire Traitement et Communication de l'Information
6 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique - ENS Paris, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
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 highdimensional 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.
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https://hal.telecom-paris.fr/hal-03494781
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Submitted on : Tuesday, May 17, 2022 - 3:18:20 PM
Last modification on : Wednesday, June 8, 2022 - 12:50:06 PM

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  • HAL Id : hal-03494781, version 1

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Kimia Nadjahi, Alain Durmus, Pierre E. Jacob, Roland Badeau, Umut Şimşekli. Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections. 35th Conference on Neural Information Processing Systems (NeurIPS 2021), Dec 2021, En ligne, France. ⟨hal-03494781⟩

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