Multi-marginal Optimal Transport Problem Research Articles

Sufficiency of -cyclical monotonicity in a class of multi-marginal optimal transport problems

Abstract $c$ -cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

Estimating pollution spread in water networks as a Schrödinger bridge problem with partial information

Incidents where water networks are contaminated with microorganisms or pollutants can result in a large number of infected or ill persons, and it is therefore important to quickly detect, localize and estimate the spread and source of the contamination. In many of today’s water networks only limited measurements are available, but with the current internet of things trend the number of sensors is increasing and there is a need for methods that can utilize this information. Motivated by this fact, we address the problem of estimating the spread of pollution in a water network given measurements from a set of sensors. We model the water flow as a Markov chain, representing the system as a set of states where each state represents the amount of water in a specific part of the network, e.g., a pipe or a part of a pipe. Then we seek the most likely flow of the pollution given the expected water flow and the sensors observations. This is a large-scale optimization problem that can be formulated as a Schrödinger bridge problem with partial information, and we address this by exploiting the connection with the entropy regularized multimarginal optimal transport problem. The software EPANET is used to simulate the spread of pollution in the water network and will be used for testing the performance of the methodology.

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].

Monge solutions and uniqueness in multi-marginal optimal transport via graph theory

We study a multi-marginal optimal transport problem with surplus b(x1,…,xm)=∑{i,j}∈Pxi⋅xj, where P⊆Q:={{i,j}:i,j∈{1,2,...m},i≠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 [15] and the surplus ∑i=1m−1xi⋅xi+1+xm⋅x1 studied in an earlier work by the present authors [26].

A Global Optimization Approach for Multimarginal Optimal Transport Problems with Coulomb Cost

In this work, we construct a novel numerical method for solving the multi-marginal optimal transport problems with Coulomb cost. This type of optimal transport problems arises in quantum physics and plays an important role in understanding the strongly correlated quantum systems. With a Monge-like ansatz, the orginal high-dimensional problems are transferred into mathematical programmings with generalized complementarity constraints, and thus the curse of dimensionality is surmounted. However, the latter ones are themselves hard to deal with from both theoretical and practical perspective. Moreover in the presence of nonconvexity, brute-force searching for global solutions becomes prohibitive as the problem size grows large. To this end, we propose a global optimization approach for solving the nonconvex optimization problems, by exploiting an efficient proximal block coordinate descent local solver and an initialization subroutine based on hierarchical grid refinements. We provide numerical simulations on some typical physical systems to show the efficiency of our approach. The results match well with both theoretical predictions and physical intuitions, and give the first visualization of optimal transport maps for some two dimensional systems.

The exactness of the [formula omitted] penalty function for a class of mathematical programs with generalized complementarity constraints

In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relation is imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The ℓ1 penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the ℓ1 penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problems in network transportation. We first provide an instance from this class, the exactness of whose ℓ1 penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.

An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems

We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography, and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving some large-scale CMOT problems on synthetic data. The preliminary experiments on the discrete tomography problem also highlight the potential modeling capability of our model.

Low-Rank Tensor Approximations for Solving Multimarginal Optimal Transport Problems

By adding entropic regularization, multi-marginal optimal transport problems can be transformed into tensor scaling problems, which can be solved numerically using the multi-marginal Sinkhorn algorithm. The main computational bottleneck of this algorithm is the repeated evaluation of marginals. Recently, it has been suggested that this evaluation can be accelerated when the application features an underlying graphical model. In this work, we accelerate the computation further by combining the tensor network dual of the graphical model with additional low-rank approximations. We provide an example for the color transfer between several images, in which these additional low-rank approximations save more than 96% of the computation time.

Transport Type Metrics on the Space of Probability Measures Involving Singular Base Measures

We develop the theory of a metric, which we call the $$\nu $$ -based Wasserstein metric and denote by $$W_\nu $$ , on the set of probability measures $${\mathcal {P}}(X)$$ on a domain $$X \subseteq \mathbb {R}^m$$ . This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $$\nu $$ and is relevant in particular for the case when $$\nu $$ is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $$\nu $$ -based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $$\nu $$ ; we also characterize it in terms of integrations of classical Wasserstein metric between the conditional probabilities when measures are disintegrated with respect to optimal transport to $$\nu $$ , and through limits of certain multi-marginal optimal transport problems. We also introduce a class of metrics which are dual in a certain sense to $$W_\nu $$ , defined relative to a fixed based measure $$\mu $$ , on the set of measures which are absolutely continuous with respect to a second fixed based measure $$\sigma $$ . As we vary the base measure $$\nu $$ , the $$\nu $$ -based Wasserstein metric interpolates between the usual quadratic Wasserstein metric (obtained when $$\nu $$ is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when $$\nu $$ is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). When $$\nu $$ concentrates on a lower dimensional submanifold of $$\mathbb {R}^m$$ , we prove that the variational problem in the definition of the $$\nu $$ -based Wasserstein metric has a unique solution. We also establish geodesic convexity of the usual class of functionals, and of the set of source measures $$\mu $$ such that optimal transport between $$\mu $$ and $$\nu $$ satisfies a strengthening of the generalized nestedness condition introduced in McCann and Pass (Arch Ration Mech Anal 238(3):1475–1520, 2020). We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.

Unbalanced Multi-marginal Optimal Transport

Entropy-regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual and show that a unique optimal transport plan exists under mild assumptions. Furthermore, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. For cost functions decoupling according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function.

Inference With Aggregate Data in Probabilistic Graphical Models: An Optimal Transport Approach

We consider inference (filtering) problems over probabilistic graphical models with aggregate data generated by a large population of individuals. We propose a new efficient belief propagation type algorithm over tree graphs with polynomial computational complexity as well as a global convergence guarantee. This is in contrast to previous methods that either exhibit prohibitive complexity as the population grows or do not guarantee convergence. Our method is based on optimal transport, or more specifically, multimarginal optimal transport theory. In particular, we consider an inference problem with aggregate observations, that can be seen as a structured multimarginal optimal transport problem where the cost function decomposes according to the underlying graph. Consequently, the celebrated Sinkhorn/iterative scaling algorithm for multi-marginal optimal transport can be leveraged together with the standard belief propagation algorithm to establish an efficient inference scheme which we call Sinkhorn belief propagation (SBP). We further specialize the SBP algorithm to cases associated with hidden Markov models due to their significance in control and estimation. We demonstrate the performance of our algorithm on applications such as inferring population flow from aggregate observations. We also show that in the special case where the aggregate observations are in Dirac form, our algorithm naturally reduces to the standard belief propagation algorithm.

Polynomial-time algorithms for multimarginal optimal transport problems with structure

Multimarginal Optimal Transport (MOT) has attracted significant interest due to applications in machine learning, statistics, and the sciences. However, in most applications, the success of MOT is severely limited by a lack of efficient algorithms. Indeed, MOT in general requires exponential time in the number of marginals k and their support sizes n. This paper develops a general theory about what “structure” makes MOT solvable in mathrm {poly}(n,k) time. We develop a unified algorithmic framework for solving MOT in mathrm {poly}(n,k) time by characterizing the structure that different algorithms require in terms of simple variants of the dual feasibility oracle. This framework has several benefits. First, it enables us to show that the Sinkhorn algorithm, which is currently the most popular MOT algorithm, requires strictly more structure than other algorithms do to solve MOT in mathrm {poly}(n,k) time. Second, our framework makes it much simpler to develop mathrm {poly}(n,k) time algorithms for a given MOT problem. In particular, it is necessary and sufficient to (approximately) solve the dual feasibility oracle—which is much more amenable to standard algorithmic techniques. We illustrate this ease-of-use by developing mathrm {poly}(n,k)-time algorithms for three general classes of MOT cost structures: (1) graphical structure; (2) set-optimization structure; and (3) low-rank plus sparse structure. For structure (1), we recover the known result that Sinkhorn has mathrm {poly}(n,k) runtime; moreover, we provide the first mathrm {poly}(n,k) time algorithms for computing solutions that are exact and sparse. For structures (2)-(3), we give the first mathrm {poly}(n,k) time algorithms, even for approximate computation. Together, these three structures encompass many—if not most—current applications of MOT.

An Optimal Transport Approach to Personalized Federated Learning

Federated learning is a distributed machine learning paradigm, which aims to train a model using the local data of many distributed clients. A key challenge in federated learning is that the data samples across the clients may not be identically distributed. To address this challenge, personalized federated learning with the goal of tailoring the learned model to the data distribution of every individual client has been proposed. In this paper, we focus on this problem and propose a novel personalized Federated Learning scheme based on Optimal Transport (FedOT) as a learning algorithm that learns the optimal transport maps for transferring data points to a common distribution as well as the prediction model under the applied transport map. To formulate the FedOT problem, we extend the standard optimal transport task between two probability distributions to multi-marginal optimal transport problems with the goal of transporting samples from multiple distributions to a common probability domain. We then leverage the results on multi-marginal optimal transport problems to formulate FedOT as a min-max optimization problem and analyze its generalization and optimization properties. We discuss the results of several numerical experiments to evaluate the performance of FedOT under heterogeneous data distributions in federated learning problems.

Genetic Column Generation: Fast Computation of High-Dimensional Multimarginal Optimal Transport Problems

We introduce a simple, accurate, and extremely efficient method for numerically solving the multi-marginal optimal transport (MMOT) problems arising in density functional theory. The method relies on (i) the sparsity of optimal plans [for $N$ marginals discretized by $\ell$ gridpoints each, general Kantorovich plans require $\ell^N$ gridpoints but the support of optimizers is of size $O(\ell\cdot N)$ [FV18]], (ii) the method of column generation (CG) from discrete optimization which to our knowledge has not hitherto been used in MMOT, and (iii) ideas from machine learning. The well-known bottleneck in CG consists in generating new candidate columns efficiently; we prove that in our context, finding the best new column is an NP-complete problem. To overcome this bottleneck we use a genetic learning method tailormade for MMOT in which the dual state within CG plays the role of an adversary, in loose similarity to Wasserstein GANs. On a sequence of benchmark problems with up to 120 gridpoints and up to 30 marginals, our method always found the exact optimizers. Moreover, empirically the number of computational steps needed to find them appears to scale only polynomially when both $N$ and $\ell$ are simultaneously increased (while keeping their ratio fixed to mimic a thermodynamic limit of the particle system).

Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation

The Strictly Correlated Electrons (SCE) limit of the Levy–Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multi-marginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. In this work, we design a numerical method, built upon constrained overdamped Langevin processes to solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced in [A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi, Approximation of optimal transport problems with marginal moments constraints, Math. Comp. 90 (2021) 689–737; C. Villani, Optimal Transport: Old and New (Springer Science & Business Media, 2008)]) of symmetric multi-marginal optimal transport problems with Coulomb cost. Some minimizers of such relaxations can be written as discrete measures charging a low number of points belonging to a space whose dimension, in the symmetrical case, scales linearly with the number of marginal laws. We leverage the sparsity of those minimizers in the design of the numerical method and prove that there is no strict local minimizer to the resulting problem. We illustrate the performance of the proposed method by numerical examples which solves MCOT relaxations of 3D systems with up to 100 electrons.

Hardness results for Multimarginal Optimal Transport problems

Multimarginal Optimal Transport (MOT) is the problem of linear programming over joint probability distributions with fixed marginals. A key issue in many applications is the complexity of solving MOT: the linear program has exponential size in the number of marginals k and their support sizes n. A recent line of work has shown that MOT is poly(n,k)-time solvable for certain families of costs that have poly(n,k)-size implicit representations. However, it is unclear what further families of costs this line of algorithmic research can encompass. In order to understand these fundamental limitations, this paper initiates the study of intractability results for MOT.Our main technical contribution is developing a toolkit for proving NP-hardness and inapproximability results for MOT problems. This toolkit reduces proving intractability of MOT problems to proving intractability of more amenable discrete optimization problems. We demonstrate this toolkit by using it to establish the intractability of a number of MOT problems studied in the literature that have resisted previous algorithmic efforts. For instance, we provide evidence that repulsive costs make MOT intractable by showing that several such problems of interest are NP-hard to solve—even approximately.

Multi-Marginal Optimal Transport and Probabilistic Graphical Models

We study multi-marginal optimal transport problems from a probabilistic graphical model perspective. We point out an elegant connection between the two when the underlying cost for optimal transport allows a graph structure. In particular, an entropy regularized multi-marginal optimal transport is equivalent to a Bayesian marginal inference problem for probabilistic graphical models with the additional requirement that some of the marginal distributions are specified. This relation on the one hand extends the optimal transport as well as the probabilistic graphical model theories, and on the other hand leads to fast algorithms for multi-marginal optimal transport by leveraging the well-developed algorithms in Bayesian inference. Several numerical examples are provided to highlight the results.

Multimarginal Optimal Transport with a Tree-Structured Cost and the Schrödinger Bridge Problem

The optimal transport problem has recently developed into a powerful framework for various applications in estimation and control. Many of the recent advances in the theory and application of optimal transport are based on regularizing the problem with an entropy term, which connects it to the Schrödinger bridge problem and thus to stochastic optimal control. Moreover, the entropy regularization makes the otherwise computationally demanding optimal transport problem feasible even for large scale settings. This has led to an accelerated development of optimal transport based methods in a broad range of fields. Many of these applications have an underlying graph structure, for instance, information fusion and tracking problems can be described by trees. In this work we consider multimarginal optimal transport problems with a cost function that decouples according to a tree structure. The entropy regularized multimarginal optimal transport problem can be viewed as a generalization of the Schrödinger bridge problem with the same tree-structure, and by utilizing these connections we extend the computational methods for the classical optimal transport problem in order to solve structured multimarginal optimal transport problems in an efficient manner. In particular, the algorithm requires only matrix-vector multiplications of relatively small dimensions. We show that the multimarginal regularization introduces less diffusion, compared to the commonly used pairwise regularization, and is therefore more suitable for many applications. Numerical examples illustrate this, and we finally apply the proposed framework for the tracking of an ensemble of indistinguishable agents.

Statistical Learning in Wasserstein Space

We seek a generalization of regression and principle component analysis (PCA) in a metric space where data points are distributions metrized by the Wasserstein metric. We recast these analyses as multimarginal optimal transport problems. The particular formulation allows efficient computation, ensures existence of optimal solutions, and admits a probabilistic interpretation over the space of paths (line segments). Application of the theory to the interpolation of empirical distributions, images, power spectra, as well as assessing uncertainty in experimental designs, is envisioned.