Related papers: Amenability, Optimal Transport and Abstract Ergodi…

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness…

Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $\partial X$. We define the critical…

We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $\Lambda$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with…

We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplica-tive ergodic theorem with an…

Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of…

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are…

The empirical optimal transport (OT) cost between two probability measures from random data is a fundamental quantity in transport based data analysis. In this work, we derive novel guarantees for its convergence rate when the involved…

We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we…

We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected…

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the…

We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $\sigma$-compact locally compact Hausdorff spaces. As its application, we prove an extension theorem of proper…

Given a determinate (multivariate) probability measure $\mu$, we characterize Gaussian mixtures $\nu\_\phi$ which minimize the Wasserstein distance $W\_2(\mu,\nu\_\phi)$ to $\mu$ when the mixing probability measure $\phi$ on the parameters…

In this paper we consider various problems involving the action of a reductive group $G$ on an affine variety $V$. We prove some general rationality results about the $G$-orbits in $V$. In addition, we extend fundamental results of Kempf…

We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the…

Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical…

We consider probability measures on $\mathbb{R}^{\infty}$ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric…

Let T be a free ergodic measure-preserving action of an abelian group G on (X,mu). The crossed product algebra R_T has two distinguished masas, the image C_T of L^infty(X,mu) and the algebra S_T generated by the image of G. We conjecture…

Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups CR satisfying the following property: given any groups G_1, G_2 in CR, then any free, ergodic, measure preserving…

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…

This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is…