Related papers: The Monge-Amp\`ere equation and its link to optima…
We give new computable necessary conditions for a class of optimal transportation problems to have smooth solutions.
For the Monge-Amp\`ere equation with a right-hand side bounded away from 0 and infinity, we show that the solution, subject to the natural boundary condition arising in optimal transport, is in $W^{2,1+\varepsilon}$ up to the boundary.
In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to…
In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by…
Mathematicians have been proposing for sometimes that Monge-Amp\`ere equation, a nonlinear generalization of the Poisson equation, where trace of the Hessian is replaced by its determinant, provides an alternative non-relativistic…
In this paper, we explore the fundamental role of the Monge-Amp\`ere equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation…
This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a…
We show that existence and uniqueness of solutions to transported Monge-Ampere problem on complex compact toric manifold follows easily from the real theory of optimal transportation.
In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the…
In this paper, we consider the global regularity for Monge-Amp\`ere type equations with the Neumann boundary conditions on Riemannian manifolds. It is known that the classical solvability of the Neumann boundary value problem is obtained…
Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi$ be the optimal transportation mapping pushing forward $\mu$ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$. In this paper, we introduce a new…
We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Amp\`ere equation,…
In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex…
Inspired by constructions in complex geometry we introduce a thermodynamic framework for Monge-Amp\`ere equations on real tori. We show convergence in law of the associated point processes and explain connections to complex Monge-Amp\`ere…
Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally…
We discuss equations associated to Cournot-Nash Equilibria as put forward recently by Blanchet and Carlier. These equations are related to an optimal transport problem in which the source measure is known, but the target measure is part of…
In this paper, we establish the global $C^{2,\alpha}$ and $W^{2,p}$ regularity for the Monge-Amp\`ere equation $\det\,D^2u = f$ subject to boundary condition $Du(\Omega) = \Omega^*$, where $\Omega$ and $\Omega^*$ are bounded convex domains…
These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures…
We give a new probabilistic construction of solutions to real Monge-Amp\`ere equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal…
We give a new proof for the interior regularity of strictly convex solutions of the Monge-Amp\`ere equation. Our approach uses a doubling inequality for the Hessian in terms of the extrinsic distance function on the maximal Lagrangian…