A User’s Guide to Optimal Transport

Abstract

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.

Distribution functions, extremal limits and optimal transport

“…The presented results on existence of optimizers and the dual problem formulation are stated in an adequate depth such that the number theoretic questions under consideration are covered. Comprehensive accounts on the subject are Villani [49] or Rachev & Rüschendorf [37] for example, a set of lecture notes on the topic would be Ambrosio & Gigli [2]. For the basic presentation of the optimal transport problem we are following the above mentioned references.…”

“…In the following sections we will apply Theorem 3.4 to problems in the spirit of [23] and in particular to problem (2). Therefore in these applications we choose X = Y = [0, 1], fix the marginal distributions to be uniform µ = ν = U[0, 1] and choose the cost function c : [0, 1] 2 → R to be continuous.…”

“…Such a measure exhibits a bi-variate distribution function C : [0, 1] 2 → [0, 1] which is generally called a copula. The aim of this article is to provide a connection between the problem of finding extremal limits in (1) (or maximal and minimal bounds for such limits) by studying the optimization problem c(x, y)γ(dx, dy) → max (2) and the field of optimal transport. Indeed, we will show how this problem can be perfectly embedded in the general theory of optimal transport.…”

“…This last problem is not easy to handle with and, apart from the already mentioned papers [23,35], only [22] is known, where the authors found an explicit asymptotic distribution function of the sequence (φ b (n), φ b (n + 1), φ b (n + 2)) n≥0 . Problem (2) has been recently studied in [25] in connection with a well-known problem in combinatorial optimization, namely the linear assignment problem. By means of this tool, the authors give optimal upper and lower bounds for integrals of two-dimensional, piecewise constant functions with respect to copulas and construct the copulas for which these bounds are attained.…”

A Sparse Multiscale Algorithm for Dense Optimal Transport

“…Over the past decades it has been studied in great detail (see for example [36] for a comprehensive monograph and some historical context and also [3,30] for helpful introductions to the subject). An important step was the polar factorization theorem [11] on R n for the cost being the squared Euclidean distance.…”