On discontinuity of planar optimal transport maps

Consider two bounded domains Ω and Λ in ℝ2, and two sufficiently regular probability measures μ and ν supported on them. By Brenier's theorem, there exists a unique transportation map T satisfying T#μ = ν and minimizing the quadratic cost ∫ℝn ∣T(x) - x∣2 dμ(x). Furthermore, by Caffarelli's regularity theory for the real Monge–Ampère equation, if Λ is convex, T is continuous. We study the reverse problem, namely, when is T discontinuous if Λ fails to be convex? We prove a result guaranteeing the discontinuity of T in terms of the geometries of Λ and Ω in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of ∂Λ to distinguish between Brenier and Alexandrov weak solutions of the Monge–Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

Download Full-text

Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

International Mathematics Research Notices ◽

10.1093/imrn/rnaa355 ◽

2020 ◽

Author(s):

Mohit Bansil ◽

Jun Kitagawa

Keyword(s):

Optimal Transport ◽

Convex Geometry ◽

Regularity Theory ◽

Uniform Norm ◽

Quantitative Stability ◽

Wirtinger Inequality ◽

Dual Variables ◽

Transport Problems ◽

Monge Ampere ◽

Stability Results

Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.

Download Full-text

Second order stability for the Monge–Ampère equation and strong Sobolev convergence of optimal transport maps

Analysis & PDE ◽

10.2140/apde.2013.6.993 ◽

2013 ◽

Vol 6 (4) ◽

pp. 993-1000 ◽

Cited By ~ 5

Author(s):

Guido De Philippis ◽

Alessio Figalli

Keyword(s):

Optimal Transport ◽

Second Order ◽

Transport Maps ◽

Monge Ampere

Download Full-text

Pseudo-Riemannian geometry encodes information geometry in optimal transport

Information Geometry ◽

10.1007/s41884-021-00053-7 ◽

2021 ◽

Author(s):

Ting-Kam Leonard Wong ◽

Jiaowen Yang

Keyword(s):

Optimal Transport ◽

Probability Distributions ◽

General Relation ◽

Regularity Theory ◽

Information Geometry ◽

Dual Pair ◽

Geometric Interpretation ◽

Primal Dual ◽

Transport Maps ◽

Riemannian Framework

AbstractOptimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate invariant properties of statistical inference. Their relations and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential-geometric relation between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, which provides a geometric perspective on the fundamental Ma–Trudinger–Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the framework of c-divergence under which divergences are defined by optimal transport maps. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor on the graph of the transport map. This relation sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the logarithmic $$L^{(\alpha )}$$ L ( α ) -divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.

Download Full-text

Characterization of barycenters in the Wasserstein space by averaging optimal transport maps

ESAIM Probability and Statistics ◽

10.1051/ps/2017020 ◽

2018 ◽

Vol 22 ◽

pp. 35-57 ◽

Cited By ~ 6

Author(s):

Jérémie Bigot ◽

Thierry Klein

Keyword(s):

Optimal Transport ◽

Probability Measures ◽

Reference Measure ◽

Precise Characterization ◽

Fixed Reference ◽

Random Probability ◽

Wasserstein Space ◽

Transport Maps ◽

Random Probability Measures

This paper is concerned by the study of barycenters for random probability measures in the Wasserstein space. Using a duality argument, we give a precise characterization of the population barycenter for various parametric classes of random probability measures with compact support. In particular, we make a connection between averaging in the Wasserstein space as introduced in Agueh and Carlier [SIAM J. Math. Anal. 43 (2011) 904–924], and taking the expectation of optimal transport maps with respect to a fixed reference measure. We also discuss the usefulness of this approach in statistics for the analysis of deformable models in signal and image processing. In this setting, the problem of estimating a population barycenter from n independent and identically distributed random probability measures is also considered.

Download Full-text

Sharp boundary ε-regularity of optimal transport maps

Advances in Mathematics ◽

10.1016/j.aim.2021.107603 ◽

2021 ◽

Vol 381 ◽

pp. 107603

Author(s):

Tatsuya Miura ◽

Felix Otto

Keyword(s):

Optimal Transport ◽

Sharp Boundary ◽

Transport Maps

Download Full-text

Regularity of optimal transport maps on multiple products of spheres

Journal of the European Mathematical Society ◽

10.4171/jems/388 ◽

2013 ◽

Vol 15 (4) ◽

pp. 1131-1166 ◽

Cited By ~ 15

Author(s):

Alessio Figalli ◽

Young-Heon Kim ◽

Robert McCann

Keyword(s):

Optimal Transport ◽

Multiple Products ◽

Transport Maps ◽

Products Of Spheres

Download Full-text

Schrödinger dynamics and optimal transport of measures

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025721500168 ◽

2021 ◽

pp. 2150016

Author(s):

Lorenzo Zanelli

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Transport Theory ◽

Optimal Transport ◽

Time Dependent ◽

Probability Measures ◽

Flat Torus ◽

Optimal Transport Theory

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

Download Full-text