Related papers: Global Regularity Estimates for Optimal Transport …
We show that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems provided that a parameter is chosen accordingly to the smoothness of the solution. This result is proven both for an a…
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, \alpha}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem…
This paper provides a new regularization method which is particularly suitable for linear exponentially ill-posed problems. Under logarithmic source conditions (which have a natural interpretation in terms of Sobolev spaces in the…
In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical…
Distributionally robust optimization has been shown to offer a principled way to regularize learning models. In this paper, we find that Tikhonov regularization is distributionally robust in an optimal transport sense (i.e., if an adversary…
The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling $\pi_{\varepsilon}$ has sparse support for small regularization parameter $\varepsilon$, in contrast to entropic…
We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity…
We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable…
This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the so-called Br\"odinger…
We propose new variational principles for traffic assignment problems. So to find equillibrium we have to solve large-scale convex optimization problem of special type. We propose some kind of "algebra" on different models and corresponding…
A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle…
We derive limit distributions for certain empirical regularized optimal transport distances between probability distributions supported on a finite metric space and show consistency of the (naive) bootstrap. In particular, we prove that the…
We establish the validity of asymptotic limits for the general transportation problem between random i.i.d. points and their common distribution, with respect to the squared Euclidean distance cost, in any dimension larger than three.…
Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, there was a lack of understanding of parabolic problems with…
Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. For instance, OT is a popular loss function that quantifies the discrepancy between an…
We propose a novel neural algorithm for the fundamental problem of computing the entropic optimal transport (EOT) plan between continuous probability distributions which are accessible by samples. Our algorithm is based on the saddle point…
A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the…
This article has two purposes. The first is to prove solutions of the second boundary value problem for generated Jacobian equations are strictly $g$-convex. The second is to prove the global $C^3$ regularity of Aleksandrov solutions to the…
In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.
Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability…