scholarly journals The geometry of optimal transportation

1996 ◽  
Vol 177 (2) ◽  
pp. 113-161 ◽  
Author(s):  
Wilfrid Gangbo ◽  
Robert J. McCann
Keyword(s):  
Optimal Transportation

scholarly journals ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS

2021 ◽  
pp. 1-37
Author(s):  
Florian F. Gunsilius
Keyword(s):  
Convergence Rate ◽  
Poincaré Inequality ◽  
Optimal Transportation ◽  
Matching Theory ◽  
Economic Research ◽  
Poincare Inequality ◽  
Optimal Transportation Problem ◽  
Minimax Rate ◽  
The Cost ◽  
Kantorovich Problem

The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

Optimal transportation plans with escort entropy regularization

2021 ◽  
Author(s):  
Takashi Kurose ◽  
Shintaro Yoshizawa ◽  
Shun-ichi Amari
Keyword(s):  
Optimal Transportation

scholarly journals Multi-physics optimal transportation and image interpolation

2015 ◽  
Vol 49 (6) ◽  
pp. 1671-1692 ◽  
Author(s):  
Romain Hug ◽  
Emmanuel Maitre ◽  
Nicolas Papadakis
Keyword(s):  
Image Interpolation ◽  
Optimal Transportation

scholarly journals Optimal transportation of flows and Nonlinear Electromagnetism

2008 ◽  
Vol 22 ◽  
pp. 15-20
Author(s):  
Yann Brenier
Keyword(s):  
Optimal Transportation
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