Related papers: An Egorov Theorem for Wasserstein Distances
Since the introduction of the Sliced Wasserstein distance in the literature, its simplicity and efficiency have made it one of the most interesting surrogate for the Wasserstein distance in image processing and machine learning. However,…
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general…
We introduce folded optimal transport, as a method to extend a cost or distance defined on the extreme boundary of a convex to the whole convex, related to convex extension. This construction broadens the framework of standard optimal…
We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $\mu$ and the corresponding empirical measure $\mu_N$, whenever $\mu$ has finite $q$-th moment for some…
Gromov-Wasserstein distances are generalization of Wasserstein distances, which are invariant under distance preserving transformations. Although a simplified version of optimal transport in Wasserstein spaces, called linear optimal…
Piecewise $\alpha$-stable Ornstein-Uhlenbeck (OU) processes arising in queue networks usually do not have an explicit dissipation, which makes the related numerical methods such as Euler-Maruyama (EM) scheme more difficult to analyze. We…
Quantum functional inequalities (e.g. the logarithmic Sobolev- and Poincar\'e inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these…
In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We…
In the history of quantum mechanics, various types of uncertainty relationships have been introduced to accommodate different operational meanings of Heisenberg uncertainty principle. We derive an optimized entropic uncertainty relation…
Based on a new Kantorovich-Rubinstein duality principle for the Hessian that was recently established by the two authors, we extend the Rio inequality to any dimension $d \ge 1$ with an optimal constant. Similarly, we propose an optimal…
In this paper we study the BV regularity for solutions of variational problems in Optimal Transportation. As an application we recover BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation…
Gromov-Wasserstein distance has found many applications in machine learning due to its ability to compare measures across metric spaces and its invariance to isometric transformations. However, in certain applications, this invariance…
Optimal transport and the Wasserstein distance $\mathcal{W}_p$ have recently seen a number of applications in the fields of statistics, machine learning, data science, and the physical sciences. These applications are however severely…
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge. Recent work has advocated a two-step approach to improve robustness and facilitate the computation of optimal transport, using…
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…
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative…
In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance…
Distinguishing quantum states with minimal sampling overhead is of fundamental importance to teach quantum data to an algorithm. Recently, the quantum Wasserstein distance emerged from the theory of quantum optimal transport as a promising…
This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal…