Related papers: Universal vector and matrix optimal transport
In recent work arXiv:2109.07820 we have shown the equivalence of the widely used nonconvex (generalized) branched transport problem with a shape optimization problem of a street or railroad network, known as (generalized) urban planning…
We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the…
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,…
This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on…
We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term,…
Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal…
We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and…
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 study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that…
A Gaussian beam method is presented for the analysis of the energy of the high frequency solution to the mixed problem of the scalar wave equation in an open and convex subset, with initial conditions compactly supported in this set, and…
Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved…
The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when…
This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh),…
We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density…
A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures $\mu$ on $\R^d$ which can be expressed as the moment measures of suitable convex functions $u$, i.e. are of the form $(\nabla u)\_\\#e^{- u}$ for…
In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized…
We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a…
We present a general (i.e., independent of the underlying model) interpolation technique based on optimal transportation of Gaussian models for parametric advection-dominated problems. The approach relies on a scalar testing function to…
We propose a new metric between probability measures on a compact metric space that mirrors the Riemannian manifold-like structure of quadratic optimal transport but includes entropic regularization. Its metric tensor is given by the…
Using the formalism of noncommutative geometric gauge theory based on the superconnection concept, we construct a new type of vector gauge theory possessing a shift-like symmetry and the usual gauge symmetry. The new shift-like symmetry is…