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

Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.

Optimal Transport for Applied Mathematicians : Calculus of Variations, PDEs, and Modeling

Transport-based techniques for signal and data analysis have recently received increased interest. Given their ability to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications, including content-based retrieval, cancer detection, image superresolution, and statistical machine learning, to name a few, and they have been shown to produce state-of-the-art results. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here, we provide a practical overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications. Software accompanying this article is available from [43].

/pdf/optimal-mass-transport-signal-processing-and-machine-2bcii5qo3a.pdf

Optimal Mass Transport: Signal processing and machine-learning applications

Scaling algorithms for unbalanced optimal transport problems

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savare. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise desciption of the Jordan-Kinderleher-Otto scheme, with proof of convergence in the easiest case: the linear Fokker-Planck equation. A discussion of other gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savare, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

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{ Euclidean, Metric, and Wasserstein } Gradient Flows: an overview

This paper introduces a numerical algorithm to compute the L2 optimal transport map between two measures µ and ν, where µ derives from a density ρ defined as a piecewise linear function (supported by a tetrahedral mesh), and where ν is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting, Aurenhammer et.al [8th Symposium on Computational Geometry conf. proc., ACM (1992)] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure µ. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses. Resume. Cet article decrit un algorithme numerique pour calculer l'application de transport optimal L2 entre deux mesures µ et ν, o` u µ derive d'une densite ρ lineaire par morceaux (supportee par un maillage tetraedrique), et o` u ν est une somme de masses de Dirac. Je donne tout d'abord une presentation elementaire de quelques resultats connus sur le transport optimal , et observe ensuite une relation avec un autreprobi eme (l'´ echantillonage optimal). Cette relation fournit des arguments simples pour etudier les fonctions objectifs caracterisant les deuxprobi emes. Je propose ensuite un algorithme pratique pour calculer le transport optimal entre une densite lineaire par morceaux et une somme de masses de Dirac en 3D. Dans ce cas semi-discret, Auren-hammer et.al [8th Symposium on Computational Geometry conf. proc., ACM (1992)] ont montre que l'application de transport optimal est determinee par les poids d'un diagramme de puissance. Les poids optimaux sont calcules en minimisant une fonction objectif convexe a l'aide d'une methode quasi-Newton. Pour evaluer cette fonction objectif et son gradient, je propose un algorithme efficace et robuste, qui calcule a chaque iteration l'intersection entre un diagramme de puissance et le maillage tetrahedrique qui definit la mesure µ. L'algorithme numerique est experimente et evalue sur plusieurs jeux de donnees , comportant jusqu'` a plusieurs centaines de milliers detetra edres et un million de masses de Dirac.

/pdf/a-numerical-algorithm-for-l-2-optimal-transport-in-3d-1b4tm5ko4d.pdf

A Numerical Algorithm for L-2 Optimal Transport in 3D

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A Numerical Algorithm for L-2 Optimal Transport in 3D