Related papers: Optimal transport and Perelman's reduced volume
In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates…
In this paper, we establish some sharp inequalities between the volume and the integral of the $k$-th mean curvature for $k+1$-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for…
In this paper we study two basic facts of optimal transportation on Wiener space W. Our first aim is to answer to the Monge Problem on the Wiener space endowed with the Sobolev type norm (k,gamma) to the power of p (cases p = 1 and p > 1…
We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter…
Many problems in geometric optics or convex geometry can be recast as optimal transport problems: this includes the far-field reflector problem, Alexandrov's curvature prescription problem, etc. A popular way to solve these problems…
We consider the entropic regularization of discretized optimal transport and propose to solve its optimality conditions via a logarithmic Newton iteration. We show a quadratic convergence rate and validate numerically that the method…
Entropically regularized optimal transport between probability measures supported on compact subsets of Euclidean space admits a representation as an information projection under moment inequality constraints. Exploiting this structure, I…
Let $M,N$ be two smooth compact hypersurfaces of $\mathbb{R}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $\mu$ and $\nu$ (with respect to the volume measures of $M$ and $N$). We consider the…
Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the…
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 study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c(x, y) = |y -- x|. The selected plan at the limit is, among those which are optimal for the non-penalized problem,…
We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper…
We study the equivalence between the weighted least gradient problem and the weighted Beckmann minimal flow problem or equivalently, the optimal transport problem with Riemannian cost. Thanks to this equivalence, we prove existence and…
We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose…
We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and propose a generalized Sinkhorn algorithm. Our framework…
This survey has been written in occasion of the School and Workshop about Optimal Transport on Quantum Structures at Erd\"os Center in September 2022. We discuss some recent results on noncommutative entropic optimal transport problems and…
Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is…
In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is…
We study the small-regularisation limit of the entropic optimal transport problem on the line with distance cost. While convergence of entropic minimizers is well understood in the discrete setting and in the case where the cost is…
We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be…