Displacement convexity of Entropy and the distance cost Optimal Transportation

Fabio Cavalletti; Nicola Gigli; Flavia Santarcangelo

doi:10.5802/afst.1679

Displacement convexity of Entropy and the distance cost Optimal Transportation

Annales de la faculté des sciences de Toulouse Mathématiques ◽

10.5802/afst.1679 ◽

2021 ◽

Vol 30 (2) ◽

pp. 411-427

Author(s):

Fabio Cavalletti ◽

Nicola Gigli ◽

Flavia Santarcangelo

Keyword(s):

Optimal Transportation ◽

Displacement Convexity

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Optimal Transportation and the Falsifiability of Incompletely Specified Economic Models. Part I: Parametric Restrictions on Latent Variables

SSRN Electronic Journal ◽

10.2139/ssrn.1123824 ◽

2008 ◽

Author(s):

Alfred Galichon ◽

Marc Henry

Keyword(s):

Latent Variables ◽

Economic Models ◽

Optimal Transportation

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ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS

Econometric Theory ◽

10.1017/s0266466621000037 ◽

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.

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