Related papers: Ramified optimal transportation in geodesic metric…
We investigate the problem of pairwise multi-marginal optimal transport, that is, given a collection of probability distributions $\{P_\alpha\}$ on a Polish space $\mathcal{X}$, to find a coupling $\{X_\alpha\}$, $X_\alpha\sim P_\alpha$,…
We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite $p$th moment for $1\leq p <\infty$, called the $\textit{sliced Monge--Kantorovich metrics}$, which include the sliced…
Describing shapes by suitable measures in object segmentation, as proposed in [24], allows to combine the advantages of the representations as parametrized contours and indicator functions. The pseudo-Riemannian structure of optimal…
What is the optimal way to deform a projective hypersurface into another one? In this paper we will answer this question adopting the point of view of measure theory, introducing the optimal transport problem between complex algebraic…
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…
We present an optimal mass transport framework on the space of Gaussian mixture models, which are widely used in statistical inference. Our method leads to a natural way to compare, interpolate and average Gaussian mixture models.…
Classical regression models do not cover non-Euclidean data that reside in a general metric space, while the current literature on non-Euclidean regression by and large has focused on scenarios where either predictors or responses are…
Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform,…
We lay out the phenomenological behavior of event-shape observables evaluated by solving optimal transport problems between collider events and reference geometries -- which we name 'manifold distances' -- to provide guidance regarding…
Numerical computation of shortest paths or geodesics on curved domains, as well as the associated geodesic distance, arises in a broad range of applications across digital geometry processing, scientific computing, computer graphics, and…
We consider the problem of finding an optimal transport plan between an absolutely continuous measure $\mu$ on $\mathcal{X} \subset \mathbb{R}^d$ and a finitely supported measure $\nu$ on $\mathbb{R}^d$ when the transport cost is the…
Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment…
We show that the Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken w.r.t. the Wasserstein Riemannian metric. Here the potential is the sum of the total classical…
In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by…
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,…
Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In…
This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou--Benamou formula for the Kantorovich metric $W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow…
The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal…
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X,dN) a possibly branching geodesic space. We show that under some assumptions on the…
We consider a path-planning scenario for a mobile robot traveling in a configuration space with obstacles under the presence of stochastic disturbances. A novel path length metric is proposed on the uncertain configuration space and then…