arXiv 2203.050052022-03math.PRmath.OC

适应最优传输的计算方法

Computational methods for adapted optimal transport

Stephan Eckstein, Gudmund Pammer

备注：: Forthcoming in Annals of Applied Probability

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文研究具有因果结构的适应最优传输问题，提出线性规划逼近、熵正则化收敛性及适应版 Sinkhorn 算法。

摘要：

适应最优传输（AOT）问题是针对时间序列分布的最优传输问题，其中耦合被约束为具有时间因果结构。本文开发了用于数值求解 AOT 问题的计算工具。首先，我们证明 AOT 问题关于边际分布的扰动是稳定的，因此任意 AOT 问题都可以由线性规划序列逼近。我们进一步研究求解 AOT 问题的熵方法。我们证明，当正则化参数趋于零时，任何熵正则化 AOT 问题都收敛到相应的非正则化问题。该证明基于一种新方法——即使在非适应情形下也是如此——可方便地获得给定固定边际耦合的光滑逼近。最后，我们证明 Sinkhorn 算法适应版本的可处理性。我们对所出现的投影给出显式解，并证明该过程收敛到熵正则化 AOT 问题的最优解。

英文摘要：

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 non-adapted 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文