COMPUTATIONAL METHODS FOR ADAPTED OPTIMAL TRANSPORT

• 成果类型: Article
• 署名作者: Eckstein, Stephan; Pammer, Gudmund
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1975
• 发表日期: 2024
• 页码: 675-713
• 关键词: causal transport; CONVERGENCE; distributions

摘要

Adapted optimal transport (AOT) problems are optimal transport problems for distributions of a time series where couplings are constrained to have a temporal causal structure. In this paper, we develop computational tools for solving AOT problems numerically. First, we show that AOT problems are stable with respect to perturbations in the marginals, and thus arbitrary AOT problems can be approximated by sequences of linear programs. We further study entropic methods to solve AOT problems. We show that any entropically regularized AOT problem converges to the corresponding unregularized problem if the regularization parameter goes to zero. The proof is based on a novel method even in the nonadapted case to easily obtain smooth approximations of a given coupling with fixed marginals. Finally, we show tractability of the adapted version of Sinkhorn's algorithm. We give explicit solutions for the occurring projections and prove that the procedure converges to the optimizer of the entropic AOT problem.