Related papers: Map-compatible decomposition of transport paths
For a family of probability spaces $\{(X_k,\mathcal{B}_{X_k},\mu_k)\}_{k=1}^N$ and a cost function $c: X_1\times\cdots\times X_N\to \mathbb{R}$ we consider the Monge-Kantorovich problem \begin{align*}\tag{MK}\label{MONKANT}…
The calculation of transport profiles from experimental measurements belongs in the category of inverse problems which are known to come with issues of ill-conditioning or singularity. A reformulation of the calculation, the matricial…
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels,…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the…
Using the growing volumes of vehicle trajectory data, it becomes increasingly possible to capture time-varying and uncertain travel costs in a road network, including travel time and fuel consumption. The current paradigm represents a road…
Recent advances in large-scale optimal transport have greatly extended its application scenarios in machine learning. However, existing methods either not explicitly learn the transport map or do not support general cost function. In this…
We define a new two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family contains the classical Monge-Kantorovich…
A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were…
We study the Monge and Kantorovich transportation problems on $\mathbb{R}^{\infty}$ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal…
Solving large scale entropic optimal transport problems with the Sinkhorn algorithm remains challenging, and domain decomposition has been shown to be an efficient strategy for problems on large grids. Unbalanced optimal transport is a…
Transportation maps between probability measures are critical objects in numerous areas of mathematics and applications such as PDE, fluid mechanics, geometry, machine learning, computer science, and economics. Given a pair of source and…
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex…
We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probability measures absolutely continuous with respect to surface measure on $\partial \Omega$, with…
Many biological systems are observed through heterogeneous modalities, requiring transport models that couple dynamics across spaces while allowing mass variation. To address this challenge, we introduce Unbalanced Synchronized Optimal…
We consider a bi-criteria generalization of the pathwidth problem, where, for given integers $k,l$ and a graph $G$, we ask whether there exists a path decomposition $\cP$ of $G$ such that the width of $\cP$ is at most $k$ and the number of…
The optimal (Monge-Kantorovich) transportation problem is discussed from several points of view. The Lagrangian formulation extends the action of the {\em Lagrangian} $L(v,x,t)$ from the set of orbits in $\R^n$ to a set of measure-valued…
Classic optimal transport theory is formulated through minimizing the expected transport cost between two given distributions. We propose the framework of distorted optimal transport by minimizing a distorted expected cost, which is the…
Optimal transport (OT) is a powerful geometric and probabilistic tool for finding correspondences and measuring similarity between two distributions. Yet, its original formulation relies on the existence of a cost function between the…
The path packing problem is stated finding the maximum number of edge-disjoint paths between predefined pairs of nodes in an undirected multigraph. Such a multigraph together with predefined node pairs is often called a network.