Sliced Wasserstein gradient flows

Computational Optimal Transport

This page illustrates the Sliced-Wasserstein-Flows Research Project for the Computational Optimal Transport (MVA) of Prof. Gabriel Peyré on the paper from Antoine Liutkus, Umut Şimşekli, Szymon Majewski, Alain Durmus, and Fabian-Robert Stöter: “Sliced-wasserstein flows: Nonparametric generative modeling via optimal transport and diffusions.” The code is available at Sliced-Wasserstein Flows.

The method consists in simulating a Wasserstein gradient flow using the Sliced-Wasserstein distance as functional aiming at retrieving a target distribution from an initial distribution (easy to sample).

The work was done on low-dimensional spaces for gaussian distributions mostly:

  1. Summarized the method and the related background
  2. Analysis of the impact of the number of random directions on the unit sphere to simulate the gradient flow
  3. Tested on various distributions
Visualization of the gradient flow starting with samples from a gaussian distribution (blue) toward a torus-shaped distribution in 3 di- mensions (red). The arrow illustrates the average direction of the gradient. Both clouds have 500 points, the step-size in the gradient descent is 1 and 500 directions were used. The scale of the axis changes between different plots