Statistical properties of the rectified transport

Music City Center 

The problem of finding a transformation mapping one distribution into another is a relevant mathematical problem with several applications in physics, genomics, etc. When this transformation is assumed to be monotonic, the above problem corresponds to finding the so-called optimal transport map, for which a rich mathematical regularity theory is available and for which a recent non-parametric estimation theory has been established. These statistical results indicate that plug-in estimators of such maps converge faster than expected for Kernel density estimators, a consequence of the extra degree of smoothness of the optimal map compared to the original densities. Moreover, a central limit theorem has been established for such estimators under suitable bandwidth selection, enabling uncertainty quantification. The main drawback is that their computation is typically intractable as it relies on solving an optimal transport problem in the continuum, for which we can only obtain approximated solutions.


To deal with these issues, we propose rectified transport as an alternative to optimal transport. The rectified map (Liu et al., 2022) is a relaxation of optimal transpor

Optimal Transport

Nonparametric estimation

Nonparametric Regression

Statistical rates 

IMS