Related papers: Optimal transport of measures via autonomous vecto…
Optimal transport (OT) is a powerful geometric tool used to compare and align probability measures following the least effort principle. Despite its widespread use in machine learning (ML), OT problem still bears its computational burden,…
We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation,…
Optimal transport distances (OT) have been widely used in recent work in Machine Learning as ways to compare probability distributions. These are costly to compute when the data lives in high dimension. Recent work by Paty et al., 2019,…
An optimal transport problem on finite spaces is a linear program. Recently, a relaxation of the optimal transport problem via strictly convex functions, especially via the Kullback--Leibler divergence, sheds new light on data sciences.…
We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target…
Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the…
We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
A fundamental concept in optimal transport is c-cyclical monotonicity: it allows to link the optimality of transport plans to the geometry of their support sets. Recently, related concepts have been successfully applied in the…
We present an iterative method to efficiently solve the optimal transportation problem for a class of strictly convex costs which includes quadratic and p-power costs. Given two probability measures supported on a discrete grid with n…
This paper studies the uniqueness of solutions to the dual optimal transport problem, both qualitatively and quantitatively (bounds on the diameter of the set of optimisers). On the qualitative side, we prove that when one marginal…
We propose a simple subsampling scheme for fast randomized approximate computation of optimal transport distances. This scheme operates on a random subset of the full data and can use any exact algorithm as a black-box back-end, including…
In the semi-discrete version of Monge's problem one tries to find a transport map $T$ with minimum cost from an absolutely continuous measure $\mu$ on $\mathbb{R}^d$ to a discrete measure $\nu$ that is supported on a finite set in…
We examine optimal matchings or transport between two stationary random measures. It covers allocation from the Lebesgue measure to a point process and matching a point process to a regular (shifted) lattice. The main focus of the article…
We propose an algorithm using method of evolving junctions to solve the optimal path planning problems with piece-wise constant flow fields. In such flow fields with a convex Lagrangian in the objective function, we can prove that the…
Flow Matching (FM) method in generative modeling maps arbitrary probability distributions by constructing an interpolation between them and then learning the vector field that defines ODE for this interpolation. Recently, it was shown that…
We establish guarantees for the unique recovery of vector fields and transport maps from finite measure-valued data, yielding new insights into generative models, data-driven dynamical systems, and PDE inverse problems. In particular, we…
Optimal transport is widely used in pure and applied mathematics to find probabilistic solutions to hard combinatorial matching problems. We extend the Wasserstein metric and other elements of optimal transport from the matching of sets to…
Inspired by the construction of the F{\"o}llmer process, we construct a unit-time flow on the Euclidean space, termed the F{\"o}llmer flow, whose flow map at time 1 pushes forward a standard Gaussian measure onto a general target measure.…
This article provides numerical simulation of an optimal transport path from a single source to an atomic measure of equal total mass. We first construct an initial transport path, and then modify the path as much as possible by using both…