有界增长函数的希尔伯特投影度量与Sinkhorn算法的指数收敛性
Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm
AI总结：
针对无界设置下的熵最优传输问题，研究有界增长可积函数空间的希尔伯特投影度量松弛锥版本，证明受控衰减核下积分算子为收缩映射，应用于熵最优传输时，轻尾边缘分布下Sinkhorn算法具指数收敛性。
摘要：
受无界设置下熵最优传输问题的启发，我们研究了有界增长可积函数空间的希尔伯特投影度量版本。这些希尔伯特度量版本源于对所有非负函数锥的松弛锥，即它们包含与特定测试函数相乘时积分值非负的所有函数。我们证明，只要核的衰减到零得到控制，即使核不远离零，核积分算子对于此类度量的适当规格也是收缩映射。作为熵最优传输的应用，我们证明当边缘分布的尾部比成本函数的增长足够轻时，Sinkhorn算法具有指数收敛性。
英文摘要：
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.
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