Optimal Transport meets Probability, Statistics and Machine Learning (17w5093)

Arriving in Oaxaca, Mexico Sunday, April 30 and departing Friday May 5, 2017

Organizers

(Université Paris Dauphine)

(ENSAE)

(University of Alberta)

(University of Cambridge)

Objectives

1) Optimal transport has long standing connections to probability, which have been amplified in recent years. For example, variants of the classical optimal transport problem have arisen in connection to applications in financial mathematics (including transport problems with additional dependence constraints and martingale optimal transport, where versions with several, or even infinitely many, prescribed marginals are also of interest) and Schrödinger's problem of minimizing the relative entropy of stochastic processes with fixed initial and final laws. In addition, Wasserstein barycenters have recently been developed as a natural tool to average or interpolate among several probability measures, against the background geometry of optimal transport. This extends the celebrated notion of displacement interpolation between two measures, and has recently found many fruitful applications, in image processing, economics and statistics. Each of these problems brings forth significant new challenges, both theoretical and computational; in addition to addressing these, we have strong reasons to believe that bringing together leaders in optimal transport with experts in probability might uncover even more connections and will stimulate research on both sides.

2) The interest in multi-marginal optimal transport problems is also rapidly growing, driven in particular by its connection with density functional theory in quantum chemistry and fluid dynamics (Brenier’s generalized solutions of incompressibe Euler). Understanding the structure, regularity and sparsity properties of optimal plans for multi-marginal transport problems is a very active and challenging area of research. Fast numerical solvers yet are still to be found to address these typically very high-dimensional problems. One of our goals in gathering specialists of optimal transport (both theoretical and computational), probability and statistics in a broad sense is to better understand how, for instance, Markov Chain Monte Carlo methods could help overcome such a computational bottleneck.

3) As outlined above, OT methods are rapidly developing in statistics and econometrics, one reason-among many others being for instance that the Brenier’s map may be viewed as one of the most natural multivariate extensions of the notion of quantile. Statistical inference based on Wasserstein distances is therefore becoming more and more popular. However, there are few rigorous limit theorems (apart from the real line case) which fully justify its use and one aim of this workshop is to make progress on such delicate issues which intimately connect probability, analysis and the geometry of Wasserstein spaces.

4)Representation of datasets, classification and measurement of similarities/disparities between complex data or objects such as images or collection of histograms are ubiquitous problems in machine learning. Optimal transport based distances are used more and more frequently to address these questions. For instance in principal component analysis (PCA) one aims to approximate in the most accurate way a large cloud of points by a small dimensional manifold and geodesic Wasserstein PCA is becoming a popular tool for large collections of histograms. Another important problem is metric learning which can somehow be viewed as a sort of inverse transport problem: what can be infered on the distance between objects given an observed coupling between them? Bringing specialists of questions using optimal transport methods, it is our hope to have a clearer picture and better geometric and analytic understanding on the performances and complexity of computational OT based methods in machine learning.