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The Sliced-Wasserstein distance (SW) is being increasingly used in machinelearning applications as an alternative to the Wasserstein distance and offerssignificant computational and statistical benefits. Since it is defined as anexpectation over random projections, SW is commonly approximated by MonteCarlo. We adopt a new perspective to approximate SW by making use of theconcentration of measure phenomenon: under mild assumptions, one-dimensionalprojections of a high-dimensional random vector are approximately Gaussian.Based on this observation, we develop a simple deterministic approximation forSW. Our method does not require sampling a number of random projections, and istherefore both accurate and easy to use compared to the usual Monte Carloapproximation. We derive nonasymptotical guarantees for our approach, and showthat the approximation error goes to zero as the dimension increases, under aweak dependence condition on the data distribution. We validate our theoreticalfindings on synthetic datasets, and illustrate the proposed approximation on agenerative modeling problem.

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