Related papers: Generalized Unnormalized Optimal Transport and its…
We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is…
Optimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density $f \in L^1(\mathbb{R}^m)$ onto another one $g \in L^1(\mathbb{R}^n)$ so as…
Many problems in dynamic data driven modeling deals with distributed rather than lumped observations. In this paper, we show that the Monge-Kantorovich optimal transport theory provides a unifying framework to tackle such problems in the…
A measure theoretical approach is presented to study the Monge-Kantorovich optimal mass transport problem. This approach together with Kantorovich duality provide an effective tool to answer a long standing question about the support of…
This article describes a set of methods for quickly computing the solution to the regularized optimal transport problem. It generalizes and improves upon the widely-used iterative Bregman projections algorithm (or Sinkhorn--Knopp…
We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation.…
This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former,…
In this note, I review entropy-regularized Monge-Kantorovich problem in Optimal Transport, and derive the gradients of several popular algorithms popular in Computational Optimal Transport, including the Sinkhorn algorithms, Wasserstein…
Capacity constrained optimal transport is a variant of optimal transport, which adds extra constraints on the set of feasible couplings in the original optimal transport problem to limit the mass transported between each pair of source and…
The Monge-Kantorovich problem for the infinite Wasserstein distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and…
We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation…
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 analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic…
In this work, we show the intrinsic relations between optimal transportation and convex geometry, especially the variational approach to solve Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. This…
In this work, we develop a collection of novel methods for the entropic-regularised optimal transport problem, which are inspired by existing mirror descent interpretations of the Sinkhorn algorithm used for solving this problem. These are…
This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to…
We study Benamou's domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove…
Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation…
We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich…
The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…