Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

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Linear And Nonlinear Programming

This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.

https://hal.archives-ouvertes.fr/hal-01096124/document

Iterative Bregman Projections for Regularized Transportation Problems

The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function. Here we consider a natural but largely unexplored variant of this problem by imposing a pointwise constraint on the joint (absolutely continuous) measures: among all joint densities with fixed marginals and which are dominated by a given density, find the optimal one. For this variant, we show local non-degeneracy of the cost function implies every minimizer is extremal in the convex set of competitors, hence unique. An appendix develops rudiments of a duality theory for this problem, which allows us to compute several suggestive examples.

/pdf/optimal-transportation-with-capacity-constraints-40qt1kacc9.pdf

Optimal transportation with capacity constraints

A variant of the classical optimal transportation problem is the following: among all joint measures with fixed marginals and that are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were established by Korman and McCann. In the present article, we expose an unexpected symmetry leading to explicit examples in two and more dimensions. These are inspired in part by simulations in one dimension that display singularities and topology and in part by two further developments: the identification of all extreme points in the feasible set and an approach to uniqueness based on constructing feasible perturbations.

/pdf/insights-into-capacity-constrained-optimal-transport-4ml6p936xg.pdf

Insights into capacity-constrained optimal transport

Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density $$f \in L^1(\mathbf {R}^m)$$
 onto another one $$g \in L^1(\mathbf {R}^n)$$
 so as to optimize a cost function $$c \in L^1(\mathbf {R}^{m+n})$$
 while respecting the capacity constraints $$0\le h \le \bar{h}\in L^\infty (\mathbf {R}^{m+n})$$
 . A linear programming duality for this problem was first proposed by Levin. In this note, we prove under mild assumptions on the given data, the existence of a pair of $$L^1$$
 -functions optimizing the dual problem. Using these functions, which can be viewed as Lagrange multipliers to the marginal constraints $$f$$
 and $$g$$
 , we characterize the solution $$h$$
 of the primal problem. We expect these potentials to play a key role in any further analysis of $$h$$
 . Moreover, starting from Levin’s duality, we derive the classical Kantorovich duality for unconstrained optimal transport. In tandem with results obtained in our companion paper [Korman et al. J Convex Anal arXiv:1309.3022
 [8] (in press)], this amounts to a new and elementary proof of Kantorovich’s duality.

/pdf/dual-potentials-for-capacity-constrained-optimal-transport-3zppopxjfo.pdf

Dual potentials for capacity constrained optimal transport

An approach to linear programming duality is proposed which relies on quadratic penalization, so that the relation between solutions to the penalized primal and dual problems becomes ane. This yields a new proof of Levin’s duality theorem for capacity-constrained optimal transport as an innite-dimension al application.

/pdf/an-elementary-approach-to-linear-programming-duality-with-1id5nzrsey.pdf

Jonathan Korman

Papers

Optimal transportation with capacity constraints

Insights into capacity-constrained optimal transport

Dual potentials for capacity constrained optimal transport

An elementary approach to linear programming duality with application to capacity constrained transport

Optimal Transportation with Capacity Constraints