ASYMPTOTICS FOR SEMIDISCRETE ENTROPIC OPTIMAL TRANSPORT

Jason M. Altschuler, Jonathan Niles-Weed, Austin J. Stromme

Research output: Contribution to journalArticlepeer-review

Abstract

We compute exact second-order asymptotics for the cost of an optimal solution to the entropic optimal transport problem in the continuous-to-discrete, or semidiscrete, setting. In contrast to the discrete-discrete or continuous-continuous case, we show that the first-order term in this expansion vanishes but the second-order term does not, so that in the semidiscrete setting the difference in cost between the unregularized and the regularized solutions is quadratic in the inverse regularization parameter, with a leading constant that depends explicitly on the value of the density at the points of discontinuity of the optimal unregularized map between the measures. We develop these results by proving new pointwise convergence rates of the solutions to the dual problem, which may be of independent interest.

Original languageEnglish (US)
Pages (from-to)1718-1741
Number of pages24
JournalSIAM Journal on Mathematical Analysis
Volume54
Issue number2
DOIs
StatePublished - 2022

Keywords

  • entropic optimal transport
  • optimal transport
  • second-order asymptotics
  • semidiscrete optimal transport

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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