The Wasserstein metric or earth mover's distance (EMD) is a useful tool in statistics, machine learning and computer science with many applications to biological or medical imaging, among others. Especially in the light of increasingly complex data, the computation of these distances via optimal transport is often the limiting factor. Inspired by this challenge, a variety of new approaches to optimal transport has been proposed in recent years and along with these new methods comes the need for a meaningful comparison.
In this paper, we introduce a benchmark for discrete optimal transport, called DOTmark, which is designed to serve as a neutral collection of problems, where discrete optimal transport methods can be tested, compared to one another, and brought to their limits on large-scale instances. It consists of a variety of grayscale images, in various resolutions and classes, such as several types of randomly generated images, classical test images and real data from microscopy.
Along with the DOTmark we present a survey and a performance test for a cross section of established methods ranging from more traditional algorithms, such as the transportation simplex, to recently developed approaches, such as the shielding neighborhood method, and including also a comparison with commercial solvers.
In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as the sum of two deterministic plans. As an application, we estimate the infinity-Wasserstein distance between two discrete probability measures $μ$ and $ν$ with the $p$-Wasserstein distance, times a constant depending on $μ$, $ν$, and the fixed cost function.
We study the optimal transport problem for $d>2$ discrete measures. This is a linear programming problem on $d$-tensors. It gives a way to compute a "distance" between two sets of discrete measures. We introduce an entropic regularization term, which gives rise to a scaling of tensors. We give a variation of the celebrated Sinkhorn scaling algorithm. We show that this algorithm can be viewed as a partial minimization algorithm of a strictly convex function. Under appropriate conditions the rate of convergence is geometric and we estimate the rate. Our results are generalizations of known resul…
Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability measures are absolutely continuous (or vanish on small sets) the theory of Wasserstein barycenters is well-developed (see the seminal paper [1]). However, exact continuous computation of Wasserstein barycenters in this setting is tractable in only a small number of specialized cases. Moreover, in many applications data is given as a set of probability measu…
We propose a new evolutionary dynamics for population games with a discrete strategy set, inspired by the theory of optimal transport and Mean field games. The dynamics can be described as a Fokker-Planck equation on a discrete strategy set. The derived dynamics is the gradient flow of a free energy and the transition density equation of a Markov process. Such process provides models for the behavior of the individual players in population, which is myopic, greedy and irrational. The stability of the dynamics is governed by optimal transport metric, entropy and Fisher information.
In this paper, we consider a discrete-time stochastic control problem with uncertain initial and target states. We first discuss the connection between optimal transport and stochastic control problems of this form. Next, we formulate a linear-quadratic regulator problem where the initial and terminal states are distributed according to specified probability densities. A closed-form solution for the optimal transport map in the case of linear-time varying systems is derived, along with an algorithm for computing the optimal map. Two numerical examples pertaining to swarm deployment demonstrate…
Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and a generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems are ubiquitous in statistics, machine learning and computer vision, however, this perception has not yet received a theoretical justification. To fill this gap, we prove that computing the Wasserstein distance between a discrete probability measure supported on two points and the Lebesgue measure on the standard hypercube is already #P-hard. This insight prompts us to se…
Loosely speaking, causal transport plans are a relaxation of adapted processes in the same sense as Kantorovich transport plans extend Monge-type transport maps. The corresponding causal version of the transport problem has recently been introduced by Lassalle. Working in a discrete time setup, we establish a dynamic programming principle that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. Based on this recursive principle, we give conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal …
We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspec…
CEREMADE, Université Paris-Dauphine（CEREMADE，巴黎第九大学）；IMB, UMR 5251, Université Bordeaux 1（IMB，波尔多第一大学）；Institut Universitaire de France（法国大学研究院）
This article introduces a generalization of the discrete optimal transport, with applications to color image manipulations. This new formulation includes a relaxation of the mass conservation constraint and a regularization term. These two features are crucial for image processing tasks, which necessitate to take into account families of multimodal histograms, with large mass variation across modes. The corresponding relaxed and regularized transportation problem is the solution of a convex optimization problem. Depending on the regularization used, this minimization can be solved using standa…
In this paper we describe a fast algorithm for generating periodic RVEs of polycrystalline materials. In particular, we use the damped Newton method from semi-discrete optimal transport theory to generate 3D periodic Laguerre tessellations (or power diagrams) with cells of given volumes. Complex, polydisperse RVEs with up to 100,000 grains of prescribed volumes can be created in a few minutes on a standard laptop. The damped Newton method relies on the Hessian of the objective function, which we derive by extending recent results in semi-discrete optimal transport theory to the periodic settin…
We consider the problem of finding consistent upper price bounds and super replication strategies for exotic options, given the observation of call prices in the market. This field of research is called model-independent finance and has been introduced by Hobson 1998. Here we use the link to mass transport problems. In contrast to existing literature we assume that the marginal distributions at the two time points we consider are discrete probability distributions. This has the advantage that the optimization problems reduce to linear programs and can be solved rather easily when assuming a ge…
In this short note, we show that given a cost function $c$, any coupling $π$ of two probability measures where the second is a discrete measure can be associated to a certain bipartite graph containing a perfect matching, based on the value of the infinity transport cost $\norm{c}_{L^\infty(π)}$. This correspondence between couplings and bipartite graphs is explicitly constructed. We give two applications of this result to the $\mathcal{W}_\infty$ optimal transport problem when the target measure is discrete, the first is a condition to ensure existence of an optimal plan induced by a mapping,…
In the current book I suggest an off-road path to the subject of optimal transport. I tried to avoid prior knowledge of analysis, PDE theory and functional analysis, as much as possible. Thus I concentrate on discrete and semi-discrete cases, and always assume compactness for the underlying spaces. However, some fundamental knowledge of measure theory and convexity is unavoidable. In order to make it as self-contained as possible I included an appendix with some basic definitions and results. I believe that any graduate student in mathematics, as well as advanced undergraduate students, can re…
Optimal transport has been used extensively in resource matching to promote the efficiency of resources usages by matching sources to targets. However, it requires a significant amount of computations and storage spaces for large-scale problems. In this paper, we take a consensus-based approach to decentralize discrete optimal transport problems and develop fully distributed algorithms with alternating direction method of multipliers. We show that our algorithms guarantee certain levels of efficiency and privacy besides the distributed nature. We further derive primal and dual algorithms by ex…
Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport we provide a framework to verify global optimality of a discrete transport plan locally. This allows construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multi-scale scheme. Any existing discrete solver can be used as internal black-box.Several cost functions, including t…
We compute exact second-order asymptotics for the cost of an optimal solution to the entropic optimal transport problem in the continuous-to-discrete, or semi-discrete, setting. In contrast to the discrete-discrete or continuous-continuous case, we show that the first-order term in this expansion vanishes but the second-order term does not, so that in the semi-discrete setting the difference in cost between the unregularized and regularized solution is quadratic in the inverse regularization parameter, with a leading constant that depends explicitly on the value of the density at the points of…
Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmetry into the solution using coupled power diagrams. This symmetry is key to capturing the discontinuities of the transport map reflected in the geometry of the power cells. We design our method as a fixed-point algorithm alternating between computatio…
We propose and analyze a modified damped Newton algorithm to solve the semi-discrete optimal transport with storage fees. We prove global linear convergence for a wide range of storage fee functions, the main assumption being that each warehouse's storage costs are independent. We show that if $F$ is an arbitrary storage fee function that satisfies this independence condition then $F$ can be perturbed into a new storage fee function so that our algorithm converges. We also show that the optimizers are stable under these perturbations. Furthermore, our results come with quantitative rates.
We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The $Γ$-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter $α$ goes to infinity. The discretization of the problem is further shown to be asymptotic preserving regarding the same limit, which ensures that a numerical method can be applied uniformly and the solutions converge to the one of the OT problem automatically. Particularly, there exists a critical value, which is independent of the mesh size, such that the discrete proble…
We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in $\mathbb{R}^d$. These metrics are natural discrete counterparts to the Kantorovich metric $\mathbb{W}_2$, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric $\mathcal{W}_{\mathcal{T}}$ in terms of $\mathbb{W}_2$, as the size of the mesh $\mathcal{T}$ tends to $0$. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional…
We show quantitative stability results for the geometric "cells" arising in semi-discrete optimal transport problems. Our results show two types of stability, the first is stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. The second is stability in Hausdorff measure, under a Poincar{è}-Wirtinger inequality and a regularity assumption equivalent to the Ma-Trudinger-Wang conditions of regularity in Monge-Amp{è}re. This last result also yields stability in the uniform norm of the dual potential functions, all three st…
Computation of Optimal Transport on Discrete Metric Measure Spaces
Institute for Applied Mathematics, University of Bonn（波恩大学应用数学研究所）；Institute for Numerical Simulation, University of Bonn（波恩大学数值模拟研究所）；Institute for Applied Mathematics, University of Münster（明斯特大学应用数学研究所）
EPFL（洛桑联邦理工学院）；Singapore University of Technology and Design（新加坡科技设计大学）
We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in K even though the size of the problem description scales linearly with K. We prove that the described optimal transport problem is #P-hard even if all components of the first rando…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Maryland（马里兰大学）
In this paper we introduce a theoretical framework for semi-discrete optimization using ideas from optimal transport. Our primary motivation is in the field of deep learning, and specifically in the task of neural architecture search. With this aim in mind, we discuss the geometric and theoretical motivation for new techniques for neural architecture search (in a companion paper we show that algorithms inspired by our framework are competitive with contemporaneous methods). We introduce a Riemannian-like metric on the space of probability measures over a semi-discrete space $\mathbb{R}^d \time…
Characterizing and computing solutions to regularized semi-discrete optimal transport via an ordinary differential equation
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de mathématiques d’Orsay（奥赛数学实验室）；ParMA；Inria Saclay（Inria萨克雷中心）；Institut Universitaire de France（法兰西学院）
This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve the discretized problems. We will describe in detail the following discretizations and corresponding algorithms: the assignment problem and Bertsekas auction's algorithm; the entropic regularization and Sinkhorn-Knopp's algorithm; semi-discrete optimal transport and Oliker-Prussner or damped Newton's algorithm, and finally semi-discrete entropic regularizat…
We propose an optimal solution to a deterministic dynamic assignment problem by leveraging connections to the theory of discrete optimal transport to convert the combinatorial assignment problem into a tractable linear program. We seek to allow a multi-vehicle swarm to accomplish a dynamically changing task, for example tracking a multi-target swarm. Our approach simultaneously determines the optimal assignment and the control of the individual agents. As a result, the assignment policy accounts for the dynamics and capabilities of a heterogeneous set of agents and targets. In contrast to a ma…
This paper deals with the large-scale behaviour of dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenis…
