# Debiaser Beware: Pitfalls of Centering Regularized Transport Maps

arXiv: https://arxiv.org/abs/2202.08919 · HTML: https://mlsanity.com/paper/2202.08919/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2022.

## Built on by (most cited first, 6 of 6)

- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Multivariate Conformal Prediction using Optimal Transport](https://mlsanity.com/paper/2502.03609/) (2025)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Progressive Entropic Optimal Transport Solvers](https://mlsanity.com/paper/2406.05061/) (2024)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [GeOT: A spatially explicit framework for evaluating spatio-temporal predictions](https://mlsanity.com/paper/2410.11709/) (2024)

## Abstract (arXiv)

Estimating optimal transport (OT) maps (a.k.a. Monge maps) between two measures P and Q is a problem fraught with computational and statistical challenges. A promising approach lies in using the dual potential functions obtained when solving an entropy-regularized OT problem between samples Pn and Qn, which can be used to recover an approximately optimal map. The negentropy penalization in that scheme introduces, however, an estimation bias that grows with the regularization strength. A well-known remedy to debias such estimates, which has gained wide popularity among practitioners of regularized OT, is to center them, by subtracting auxiliary problems involving Pn and itself, as well as Qn and itself. We do prove that, under favorable conditions on P and Q, debiasing can yield better approximations to the Monge map. However, and perhaps surprisingly, we present a few cases in which debiasing is provably detrimental in a statistical sense, notably when the regularization strength is large or the number of samples is small. These claims are validated experimentally on synthetic and real datasets, and should reopen the debate on whether debiasing is needed when using entropic optimal transport.