We study a multi-marginal optimal transport problem with surplus $b(x_{1}, \ldots, x_{m})=\sum_{\{i,j\}\in P} x_{i}\cdot x_{j}$, where $P\subseteq Q:=\{\{i,j\}: i, j \in \{1,2,...m\}, i \neq j\}$. We reformulate this problem by associating each surplus of this type with a graph with $m$ vertices whose set of edges is indexed by $P$. We then establish uniqueness and Monge solution results for two general classes of surplus functions. Among many other examples, these classes encapsulate the Gangbo and Świȩch surplus [12] and the surplus $\sum_{i=1}^{m-1}x_{i}\cdot x_{i+1} + x_{m}\cdot x_{1}$ studied in an earlier work by the present authors [23].
This note establishes that a generalization of $c$-cyclical monotonicity from the Monge-Kantorovich problem with two marginals gives rise to a sufficient condition for optimality also in the multi-marginal version of that problem. To obtain the result, the cost function is assumed to be bounded by a sum of integrable functions. The proof rests on ideas from martingale transport.
We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in [23]. In addition, it is satisfied by the special cost functions in our earlier work [32, 33], when absolute continuity is imposed on certain other collections of marginals. We also pres…
We shall present a measure theoretical approach for which together with the Kantorovich duality provide an efficient tool to study the optimal transport problem. Specifically, we study the support of optimal plans where the cost function does not satisfy the classical twist condition in the two marginal problem as well as in the multi-marginal case when twistedness is limited to certain subsets.
It is known from clever mathematical examples \cite{Ca10} that the Monge ansatz may fail in continuous two-marginal optimal transport (alias optimal coupling alias optimal assignment) problems. Here we show that this effect already occurs for finite assignment problems with $N=3$ marginals, $\ell=3$ 'sites', and symmetric pairwise costs, with the values for $N$ and $\ell$ both being optimal. Our counterexample is a transparent consequence of the convex geometry of the set of symmetric Kantorovich plans for $N=\ell=3$, which -- as we show -- possess 22 extreme points, only 7 of which are Monge.…
This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational method. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the duality method gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of optimal mapping fo…
We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. These problems consist of minimizing a linear objective function over a high-dimensional polytope, here referred to as Kantorovich polytope. The presented results are of split nature, computational and theoretical. Within the computational part only small numbers of marginals $N$ and marginal sites $\ell$ are considered. This restriction allows us to computationally determine all extreme points of the Kantorovich polytope and investigate how many of them are in c…
We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution must concentrate on either finitely many or countably many graphs. We show that local differential conditions on the cost, known to imply local $d$-rectifiability of the solution, are sufficient to imply a local version of the first of our conditions. We exhibit two examples of cost functions satisfying our conditions, including the Coulomb cost from density…
It is well-known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance, and letting this parameter tend to one. A complete construction of this solution that we call excursion coupling is given. This is reminiscent to the one in the convex case. It is also characterized as the solution of secondary transport problems. Moreover, a combinatoric/geometric characterization of the routes used for this tra…
Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network based optimal transport map solver has gained great attention in recent years. Along this line, we present a scalable algorithm for computing the neural Monge map between two probability distributions. Our algorithm is based on a weak form of the optimal transport problem, thus it only requires samples from the marginals instead of their analytic expressions, and can accommodate optimal transport between two distribution…
Over the past five years, multi-marginal optimal transport, a generalization of the well known optimal transport problem of Monge and Kantorovich, has begun to attract considerable attention, due in part to a wide variety of emerging applications. Here, we survey this problem, addressing fundamental theoretical questions including the uniqueness and structure of solutions. The (partial) answers to these questions uncover a surprising divergence from the classical two marginal setting, and reflect a delicate dependence on the cost function. We go one to describe two applications of the multi-ma…
We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a…
A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.
We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Carathéodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.
We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Swiech (1998). We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results of Gangbo and Swiech. The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier (2011).
We consider the $L^\infty$-optimal mass transportation problem \[ \min_{Π(μ, ν)} γ-\mathrm{ess\,sup\,} c(x,y), \] for a new class of costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In particular we study the conditions under which the infinitely-motonone minimizers are induced by a transportation map. We also state a uniqueness result for infinitely cyclically monotone Monge minimizers that corresponds to this class of cost functions. We compare the results to previous works.
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $μ, ν$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimension $d$ is arbitrary.
In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of large deviations of the empirical distribution and an early example of the so-called maximum entropy inference method. This so-called Schroedinger bridge problem (SBP) was recently recognized as a regularization of the Monge-Kantorovich Optimal Mass Transport (OMT), leading to effective computation of the latter. Specifically, OMT with quadratic cost may be viewed as a zero-temperature limit of SBP, which amounts to minimization of the Helmholtz's free energy over probability distributions constrained to possess give…
We prove existence and uniqueness results for solutions to a class of optimal transportation problems with infinitely many marginals, supported on the real line. We also provide a characterization of the solution with an explicit formula. We then show that this result implies an infinite dimensional rearrangement inequality, and use it to derive upper bounds on solutions to parabolic PDE via the Feynman-Kac formula. This result can be used to provide model independent bounds on the prices of certain derivatives in mathematical finance. As an another application, we obtain refined phase-space b…
This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cr…
