  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.
