Entropic optimal transport: convergence of potentials

• 成果类型: Article
• 署名作者: Nutz, Marcel; Wiesel, Johannes
• 署名单位: Columbia University; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01096-8
• 发表日期: 2022
• 页码: 401-424
• 关键词: minimization

摘要

We study the potential functions that determine the optimal density for epsilon-entropically regularized optimal transport, the so-called Schrodinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit epsilon -> 0 of vanishing regularization, strong compactness holds in L-1 and cluster points are Kantorovich potentials. In particular, the Schrodinger potentials converge in L-1 to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schrodinger bridges, the limit corresponds to the small-noise regime.