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Brendan Pass

Publications and source records attributed to Brendan Pass.

At least 19 recordsLinked to original sources
arXiv2026

Ordinary differential equations for regularized variational problems involving semi-discrete optimal transport

We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed ordinary differential equations in the regularization parameter. The initial conditions for these equations, corresponding to solutions to completely regularized problems, are typically known explicitly. The ODE can then be solved to recover the solution for an arbitrary degree of regularization; we verify that the solution is continuous in the regularization parameter, implying that taking the limit of the trajectory yields the solution to the fully unregularized problem. We establish analogous results for a version of the problem when the non-optimal transport term is not scaled with the regularization parameter. We exploit our characterization to numerically solve several example problems using standard ODE methods; this strategy exhibits superior robustness to alternatives such as Newton's method, as arbitrary initializations are not required.

math.OC↗
arXiv2025

A dynamical formulation of multi-marginal optimal transport

We present a primal-dual dynamical formulation of the multi-marginal optimal transport problem for (semi-)convex cost functions. Even in the two-marginal setting, this formulation applies to cost functions not covered by the classical dynamical approach of Benamou-Brenier. Our dynamical formulation yields a convex optimization problem, enabling the use of convex optimization tools to find quasi-Monge solutions of the static multi-marginal problem for translation-invariant costs. We illustrate our results numerically with proximal splitting methods.

math.OC↗
arXiv2025

Semi-discrete multi-to -one dimensional variational problems

We study a class of semi-discrete variational problems that arise in economic matching and game theory, where agents with continuous attributes are matched to a finite set of outcomes with a one dimensional structure. Such problems appear in applications including Cournot-Nash equilibria, and hedonic pricing, and can be formulated as problems involving optimal transport between spaces of unequal dimensions. In our discrete strategy space setting, we establish analogues of results developed for a continuum of strategies in \cite{nenna2020variational}, ensuring solutions have a particularly simple structure under certain conditions. This has important numerical consequences, as it is natural to discretize when numerically computing solutions. We leverage our results to develop efficient algorithms for these problems which scale significantly better than standard optimal transport solvers, particularly when the number of discrete outcomes is large, provided our conditions are satisfied. We also establish rigorous convergence guarantees for these algorithms. We illustrate the advantages of our approach by solving a range of numerical examples; in many of these our new solvers outperform alternatives by a considerable margin.

math.OC↗
arXiv2025

Multi-to -one dimensional and semi-discrete screening

We study the monopolist's screening problem with a multi-dimensional distribution of consumers and a one-dimensional space of goods. We establish general conditions under which solutions satisfy a structural condition known as nestedness, which greatly simplifies their analysis and characterization. Under these assumptions, we go on to develop a general method to solve the problem, either in closed form or with relatively simple numerical computations, and illustrate it with examples. These results are established both when the monopolist has access to only a discrete subset of the one-dimensional space of products, as well as when the entire continuum is available. In the former case, we also establish a uniqueness result.

math.OC↗
arXiv2025

Remarks on multi-period martingale optimal transport

We study the structural properties of multi-period martingale optimal transport (MOT). We develop new tools to address these problems, and use them to prove several uniqueness and structural results on three-period martingale optimal transport. More precisely, we establish lemmas on how and when two-period martingale couplings may be glued together to obtain multi-period martingales and which among these glueings are optimal for particular MOT problems. We use these optimality results to study limits of solutions under convergence of the cost function and obtain a corresponding linearization of the optimal cost. We go on to establish a complete characterization of limiting solutions in a three-period problem as the interaction between two of the variables vanishes. Under additional assumptions, we show uniqueness of the solution and a structural result which yields the solution essentially explicitly. For the full three-period problem, we also obtain several structural and uniqueness results under a variety of different assumptions on the marginals and cost function. We illustrate our results with a real world application, providing approximate model independent upper and lower bounds for options depending on Amazon stock prices at three different times. We compare these bounds to prices computed using certain models.

math.OC↗
arXiv2025

Characterizing and computing solutions to regularized semi-discrete optimal transport via an ordinary differential equation

This paper investigates the semi-discrete optimal transport (OT) problem with entropic regularization. We characterize the solution using a governing, well-posed ordinary differential equation (ODE). This naturally yields an algorithm to solve the problem numerically, which we prove has desirable properties, notably including global strong convexity of a value function whose Hessian must be inverted in the numerical scheme. Extensive numerical experiments are conducted to validate our approach. We compare the solutions obtained using the ODE method with those derived from Newton's method. Our results demonstrate that the proposed algorithm is competitive for problems involving the squared Euclidean distance and exhibits superior performance when applied to various powers of the Euclidean distance. Finally, we note that the ODE approach yields an estimate on the rate of convergence of the solution as the regularization parameter vanishes, for a generic cost function.

math.NA↗
arXiv2025

Partial Identification in Moment Models with Incomplete Data--A Conditional Optimal Transport Approach

In this paper, we develop a unified approach to study partial identification of a finite-dimensional parameter defined by a general moment model with incomplete data. We establish a novel characterization of the identified set for the true parameter in terms of a continuum of inequalities defined by conditional optimal transport. For the special case of an affine moment model, we show that the identified set is convex and that its support function can be easily computed by solving a conditional optimal transport problem. For parameters that may not satisfy the moment model, we propose a two-step procedure to construct its identified set. Finally, we demonstrate the generality and effectiveness of our approach through several running examples.

econ.EM↗
arXiv2025

Data denoising with self consistency, variance maximization, and the Kantorovich dominance

We introduce a new framework for data denoising, partially inspired by martingale optimal transport. For a given noisy distribution (the data), our approach involves finding the closest distribution to it among all distributions which 1) have a particular prescribed structure (expressed by requiring they lie in a particular domain), and 2) are self-consistent with the data. We show that this amounts to maximizing the variance among measures in the domain which are dominated in convex order by the data. For particular choices of the domain, this problem and a relaxed version of it, in which the self-consistency condition is removed, are intimately related to various classical approaches to denoising. We prove that our general problem has certain desirable features: solutions exist under mild assumptions, have certain robustness properties, and, for very simple domains, coincide with solutions to the relaxed problem. We also introduce a novel relationship between distributions, termed Kantorovich dominance, which retains certain aspects of the convex order while being a weaker, more robust, and easier-to-verify condition. Building on this, we propose and analyze a new denoising problem by substituting the convex order in the previously described framework with Kantorovich dominance. We demonstrate that this revised problem shares some characteristics with the full convex order problem but offers enhanced stability, greater computational efficiency, and, in specific domains, more meaningful solutions. Finally, we present simple numerical examples illustrating solutions for both the full convex order problem and the Kantorovich dominance problem.

stat.ME↗
arXiv2024

The Monge-Kantorovich problem on Wasserstein space

We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures $\mathbb{P}_1,\mathbb{P}_2\in\mathcal{P}(\mathcal{P}(M))$ on the space $\mathcal{P}(M)$ of probability measures on a smooth compact manifold, we study the optimal transport problem between $\mathbb{P}_1$ and $\mathbb{P}_2 $ where the cost function is given by the squared Wasserstein distance $W_2^2(\mu,\nu)$ between $\mu,\nu \in \mathcal{P}(M)$. Under appropriate assumptions on $\mathbb{P}_1$, we prove that there exists a unique optimal plan and that it takes the form of an optimal map. An extension of this result to cost functions of the form $h(W_2(\mu,\nu))$, for strictly convex and strictly increasing functions $h$, is also established. The proofs rely heavily on a recent result of Schiavo \cite{schiavo2020rademacher}, which establishes a version of Rademacher's theorem on Wasserstein space.

math.PR↗
arXiv2024

An ordinary differential equation for entropic optimal transport and its linearly constrained variants

We characterize the solution to the entropically regularized optimal transport problem by a well-posed ordinary differential equation (ODE). Our approach works for discrete marginals and general cost functions, and in addition to two marginal problems, applies to multi-marginal problems and those with additional linear constraints. Solving the ODE gives a new numerical method to solve the optimal transport problem, which has the advantage of yielding the solution for all intermediate values of the ODE parameter (which is equivalent to the usual regularization parameter). We illustrate this method with several numerical simulations. The formulation of the ODE also allows one to compute derivatives of the optimal cost when the ODE parameter is $0$, corresponding to the fully regularized limit problem in which only the entropy is minimized.

math.OC↗
arXiv2024

On the existence of Monge solutions to multi-marginal optimal transport with quadratic cost and uniform discrete marginals

A natural and important question in multi-marginal optimal transport is whether the \emph{Monge ansatz} is justified; does there exist a solution of Monge, or deterministic, form? We address this question for the quadratic cost when each marginal measure is $m$-empirical (that is, uniformly supported on $m$ points). By direct computation, we provide an example showing that the ansatz \emph{can fail} when the underlying dimension $d$ is $2$, number of marginals $N$ to be matched is $3$ and the size $m$ of their supports is $3$. As a consequence, the set of $m$-empirical measures is not barycentrically convex when $N \geq 3$, $d \geq 2$ and $m \geq3$. It is a well known consequence of the Birkhoff-von Neumann Theorem that the Monge ansatz holds for $N=2$, standard techniques show it holds when $d=1$, and we provide a simple proof here that \emph{it holds whenever $m=2$}. Therefore, the $N$, $d$ and $m$ in our counterexample are as small as possible.

math.OC↗
arXiv2023

Dimension Reduction in Martingale Optimal Transport: Geometry and Robust Option Pricing

This paper addresses the problem of robust option pricing within the framework of Vectorial Martingale Optimal Transport (VMOT). We investigate the geometry of VMOT solutions for $N$-period market models and demonstrate that, when the number of underlying assets is $d=2$ and the payoff is sub- or supermodular, the extremal model reduces to a single-factor structure in the first period. This structural result allows for a significant dimension reduction, transforming the problem into a more tractable format. We prove that this reduction is specific to the two-asset case and provide counterexamples showing it generally fails for $d \geq 3$. Finally, we exploit this monotonicity to develop a reduced-dimension Sinkhorn algorithm. Numerical experiments demonstrate that this structure-preserving approach reduces computational time by approximately 99\% compared to standard methods while improving accuracy.

q-fin.MF↗
arXiv2022

An ODE characterisation of multi-marginal optimal transport with pairwise cost functions

The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one parameter family of cost functions that interpolates between the original and a special cost function for which the problem's complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $\epsilon$, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge-Kutta schemes to compute solutions to the ODE, and, as a result, the multi-marginal optimal transport problem.

math.OC↗
arXiv2022

Robust risk management via multi-marginal optimal transport

We study the problem of maximizing a spectral risk measure of a given output function which depends on several underlying variables, whose individual distributions are known but whose joint distribution is not. We establish and exploit an equivalence between this problem and a multi-marginal optimal transport problem. We use this reformulation to establish explicit, closed form solutions when the underlying variables are one dimensional, for a large class of output functions. For higher dimensional underlying variables, we identify conditions on the output function and marginal distributions under which solutions concentrate on graphs over the first variable and are unique, and, for general output functions, we find upper bounds on the dimension of the support of the solution. We also establish a stability result on the maximal value and maximizing joint distributions when the output function, marginal distributions and spectral function are perturbed; in addition, when the variables one dimensional, we show that the optimal value exhibits Lipschitz dependence on the marginal distributions for a certain class of output functions. Finally, we show that the equivalence to a multi-marginal optimal transport problem extends to maximal correlation measures of multi-dimensional risks; in this setting, we again establish conditions under which the solution concentrates on a graph over the first marginal.

math.OC↗
arXiv2022

Lorenz map, inequality ordering and curves based on multidimensional rearrangements

We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022.

econ.EM↗
arXiv2022

Monge solutions and uniqueness in multi-marginal optimal transport: weaker conditions on the cost, stronger conditions on the marginals

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 present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals.

math.OC↗
arXiv2022

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 distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure $\nu$, the $\nu$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when $\nu$ is sufficiently regular. 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 distance has a unique solution. We 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 \cite{McCannPass20}.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.

math.OC↗