Related papers: On relations between transportation cost spaces an…
The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schr\"odinger map. We prove that when the cost function is $\mathcal{C}^{k+1}$ with $k\in \mathbb{N}^*$…
Let $M$, $N$ be finite-dimensional manifolds with $M$ compact. This paper looks at the Riemnannian geometry on the space $C^\infty(M,N)$ of smooth maps equipped with the $L^2$-Riemannian metric. This metric was used by Ebin and Marsden in…
Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…
We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant…
We study multi-marginal optimal transport problems from a probabilistic graphical model perspective. We point out an elegant connection between the two when the underlying cost for optimal transport allows a graph structure. In particular,…
We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to $\ell_1$. This extends results of A. Dalet using an alternative approach.
This paper introduces the first statistically consistent estimator of the optimal transport map between two probability distributions, based on neural networks. Building on theoretical and practical advances in the field of Lipschitz neural…
Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $M$ be a complete Riemnnian manifold and…
Let $ G $ be a simple graph of $ \ell $ vertices $ \{1, \dots, \ell \} $ with edge set $ E_{G} $. The graphical arrangement $ \mathcal{A}_{G} $ consists of hyperplanes $ \{x_{i}-x_{j}=0\} $, where $ \{i, j \} \in E_{G} $. It is well known…
The goal of this paper is to study geometric and extremal properties of the convex body $B_{\mathcal F(M)}$, which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space $M$. We investigate $\ell_1$ and…
During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott-Sturm-Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in…
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…
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 solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space $M$, the set consisting of Lipschitz functions on $M$ which do not strongly attain their norm…
In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as…
We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight…
We investigate the optimal mass transport problem associated to the following "ballistic" cost functional on phase space $M\times M^*$, $$ b_T(v, x):=\inf\{\langle v, \gamma (0)\rangle +\int_0^TL(\gamma (t), {\dot \gamma}(t))\, dt, \gamma…
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…
Metro networks serve as good examples of traffic systems for understanding the relations between geometric structures and transport properties.We study and compare 28 world major metro networks in terms of the Wasserstein distance, the key…