Debiaser Beware: Pitfalls of Centering Regularized Transport Maps

Aram Alexandre Pooladian, Marco Cuturi, Jonathan Niles-Weed

Research output: Contribution to journalConference articlepeer-review


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 OT.

Original languageEnglish (US)
Pages (from-to)17830-17847
Number of pages18
JournalProceedings of Machine Learning Research
StatePublished - 2022
Event39th International Conference on Machine Learning, ICML 2022 - Baltimore, United States
Duration: Jul 17 2022Jul 23 2022

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability


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