Appears in collection : 2022 - T3 - WS3 - Measure-theoretic Approaches and Optimal Transportation in Statistics

Optimal transport is a fundamental tool to deal with discrete and continuous distributions of points. We can understand it either as a generalization of sorting to spaces of dimension D sup 1, or as a nearest neighbor projection under an incompressibility constraint. Over the last decade, a sustained research effort on numerical foundations has led to a x1,000 speed-up for most transport-related computations. Computing Earth Mover’s Distances or Wasserstein barycenters between 3D volumes and surfaces is now a matter of milliseconds. This has opened up a wide range of research directions in geometric data analysis, machine learning and computer graphics. This lecture will discuss the consequences of these game-changing numerical advances from a user’s perspective. We will focus on: 1. Mature libraries and software tools that can be used as of 2022, with a clear picture of the current state-of-the-art. 2. New ranges of applications in 3D shape analysis, with a focus on population analysis and shape registration. 3. Open problems that remain to be solved by experts in the field.