Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm

• 成果类型: Article
• 署名作者: Eckstein, Stephan
• 署名单位: Eberhard Karls University of Tubingen
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01366-9
• 发表日期: 2025
• 页码: 585-621
• 关键词: optimal transport; frobenius theory; minimization; THEOREMS; matrices; geometry

摘要

Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert's metric originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function.