相关论文: On the transport dimension of measures
We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost…
The optimal transportation problem, first suggested by Gaspard Monge in the 18th century and later revived in the 1940s by Leonid Kantorovich, deals with the question of transporting a certain measure to another, using transport maps or…
In this note, we propose an extension of the Wasserstein 1-metric ($W_1$) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular,…
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…
Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (\`a la Benamou-Brenier) to static…
We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field…
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…
We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric…
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…
This article generalizes the study of branched/ramified optimal transportation to those with capacity constraints. Each admissible transport network studied here is represented by a transport multi-path between measures, with a capacity…
Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this…
This paper develops a comprehensive theory of optimal transport for signed (real) measures on Rd. Extending the classical Brenier theorem, we consider Jordan decompositions of measures with possibly fractal singular parts. Under suitable…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
This paper continues the investigation of `Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N…
In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is…
We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal…
We introduce the optimal transportation interpretation of the Kantorovich norm on thespace of signed Radon measures with finite mass, based on a generalized Wasserstein distancefor measures with different masses.With the formulation and the…
Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on…
We consider an optimal transport problem between laws of random probability measures: given a base cost function, we build the associated OT cost between probability measures that in turn we use to define the OT cost between probability…
We study couplings $q^\bullet$ of two equivariant random measures $\lambda^\bullet$ and $\mu^\bullet$ on a Riemannian manifold $(M,d,m)$. Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the…