Related papers: Statistical Estimation of Monge Transport Maps via…
We present a general method, based on conjugate duality, for solving a convex minimization problem without assuming unnecessary topological restrictions on the constraint set. It leads to dual equalities and characterizations of the…
We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex…
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…
We present a deep generative model, named Monge-Amp\`ere flow, which builds on continuous-time gradient flow arising from the Monge-Amp\`ere equation in optimal transport theory. The generative map from the latent space to the data space…
We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each…
Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of…
We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier. Our extension allows optimal transfer of unnormalized and unequal masses. We…
In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action…
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal…
We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex…
Given a source and a target probability measure supported on $\mathbb{R}^d$, the Monge problem asks to find the most efficient way to map one distribution to the other. This efficiency is quantified by defining a \textit{cost} function…
Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semi-discrete or fully-discrete optimal transport problems. These two problems ensue from approximating $\mu$ or both…
We consider Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge…
We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…
A numerical method for the solution of the elliptic Monge-Ampere Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation (OT) problem is presented. A local representation of the OT boundary…
We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on…
We propose a numerical method for solving the multi-marginal Monge problem, which extends the classical Monge formulation to settings involving multiple target distributions. Our approach is based on the Hilbert space embedding of…
Quasi-Monte Carlo (QMC) is a powerful method for evaluating high-dimensional integrals. However, its use is typically limited to distributions where direct sampling is straightforward, such as the uniform distribution on the unit hypercube…
The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Amp\`ere type known as generated Jacobian equations. These equations, whose general existence theory has been recently…
We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb R^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption…