Abstract
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.
Highlights
Optimal transport theory has been an active area of research in applied mathematics, machine learning, statistics, and economics
This problem has the same regularity properties in multiple dimensions as the infinitesimal generators of ergodic diffusions, which has been shown to be higher than the regularity of classical smoothed empirical processes by the seminal result Dalalyan and Reiß (2007, Prop. 1), see the analysis in Rohde and Strauch (2010)
This strongly suggests that an application of these results in place of the classical results for smoothed empirical processes can lead to fewer restrictions on the admissible bandwidth which would imply the minimax rate of convergence found in Hütter and Rigollet (2019) for the simple plug-in estimator, without changing the estimator or mathematical results of this paper
Summary
Optimal transport theory has been an active area of research in applied mathematics, machine learning, statistics, and economics. This rate is slightly suboptimal compared to the minimax rate of the Brenier map derived in Hütter and Rigollet (2019) The reason for this is not our derived results of the semi-dual Monge–Kantorovich problem but is due to the fact that we rely on classical results for the rate of convergence of smoothed empirical processes in Giné and Nickl (2008) and Radulovicand Wegkamp (2000). As the main mathematical result of this article, we show that the second variation of the semi-dual Monge–Kantorovich problem takes the form of a Dirichlet energy functional weighted by the density function of the source measure This problem has the same regularity properties in multiple dimensions as the infinitesimal generators of ergodic diffusions, which has been shown to be higher than the regularity of classical smoothed empirical processes by the seminal result Dalalyan and Reiß Appendixes A and B contain a brief review of the Monge–Kantorovich problem and all proofs, including additional lemmas
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