MAC-MIGS afternoon: Computational Optimal Transport and Applications
Date and time: Friday 10 July, 13:30
Organisers: David Bourne and Beatrice Pelloni
13:30-14:30: Quentin Mérigot (Université Paris-Sud)
14:45-15:45: Aude Genevay (MIT)
16:00-17:00: Colin Cotter (Imperial College London)
Titles and abstracts:
Title: Semi-discrete optimal transport
Abstract: In this talk, we will review the semi-discrete methods for optimal transport, a class of numerical methods which has a very geometric flavor. They rely on tools similar to Voronoi tessellations, and are especially very efficient for 2D and 3D problems, for a quadratic cost. We will review the formulation of the (semi-)discretized problems and some algorithms for solving them. If time permits, we will discuss convergence of the solutions of the discrete problems to continuous ones, and some applications.
Title: Learning with entropy-regularized optimal transport
Abstract: Entropy-regularized OT (EOT) was first introduced by Cuturi in 2013 as a solution to the computational burden of OT for machine learning problems. In this talk, after studying the properties of EOT, we will introduce a new family of losses between probability measures called Sinkhorn Divergences. Based on EOT, this family of losses actually interpolates between OT (no regularization) and MMD (infinite regularization). We will illustrate these theoretical claims on a set of learning problems formulated as minimizations over the space of measures.
Title: Discrete Monge Ampere approaches to solving the semigeostrophic equations
Abstract: In this talk I will revisit the geometric algorithm for solving the semigeostrophic equations using optimal transport. The semigeostrophic equations describe atmospheric motion in a particular limit that is relevant to the problem of frontogenesis. Since their introduction by Hoskins, these equations have been understood through an interpretation as dynamics driven by an optimal transport problem. In the 1990s, Cullen, Roulstone and others worked on a geometric algorithm for solving a discretised version of this optimal transport problem. The algorithm was not taken much further as it involved a complicated direct solver algorithm that was computationally challenging. In this talk I will revisit this algorithm in the light of recent work by Mérigot et al. on iterative methods for discrete optimal transport problems, which opens the possibility of making higher resolution solutions that could be used for benchmarking more standard numerical schemes.