Related papers: Connes distance and optimal transport
We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying…
The seminal result of Benamou and Brenier provides a characterization of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is…
If M is a smooth compact connected Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P(M). We show…
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…
The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the…
In this paper we deal with the analysis of the solutions of traffic flow models at multiple scales, both in the case of a single road and of road networks. We are especially interested in measuring the distance between traffic states (as…
Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as a powerful discrepancy measure for probability distributions. The recent developments on…
Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from…
An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators…
We establish various bounds on the solutions to a Stein equation for Poisson approximation in Wasserstein distance with non-linear transportation costs. The proofs are a refinement of those in [Barbour and Xia (2006)] using the results in…
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…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
We determine the isometry group of the $n$-qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all $n \in \mathbb{N}.$ It turns out that the isometries are precisely the…
Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance. Recent work suggests that this quantity is more robust than the standard Wasserstein distance, in…
We construct spectral triples of one- and two-qubit states using the Hilbert-Schmidt operatorial formulation, and study the Connes spectral distances. We also construct the Dirac operator corresponding to the normal quantum trace distances.…
We study a class of quadratic Wasserstein distances on spaces consisting of generalized dynamical systems on a von Neumann algebra. We emphasize how symmetry of such a Wasserstein distance arises, but also study the asymmetric case. This…
We elucidate why the 1-Wasserstein distance $W_1$ coincides with the area between the two marginal cumulative distribution functions (CDFs). We first describe the Wasserstein distance in terms of copulas, and then show that $W_1$ with the…
We study the problem of quantifying how far an empirical distribution deviates from Gaussianity under the framework of optimal transport. By exploiting the cone geometry of the relative translation invariant quadratic Wasserstein space, we…
Optimal transport distances are powerful tools to compare probability distributions and have found many applications in machine learning. Yet their algorithmic complexity prevents their direct use on large scale datasets. To overcome this…
Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by…