论文标题
精确的指数，用于产品抛光空间中的浓度和等级的指数
Exact Exponents for Concentration and Isoperimetry in Product Polish Spaces
论文作者
论文摘要
在本文中，我们得出了在某些轻度假设下，在产品抛光剂概率空间中，浓度和等值函数的渐近指数（即收敛速率）的变异公式。这些公式以相对熵（来自信息理论）和最佳运输成本功能（来自最佳运输理论）表示。因此，我们的结果验证了信息理论，最佳运输以及量度或等值不平等的集中度之间的紧密联系。在集中度方案中，相应的变分公式实际上是无维的界限，因为该界限对任何维度都是有效的。提供了辅助随机变量在渐近等级指数表达中的字母的基础性，这使得通过有限字母案例的有限维程序计算的表达式可以计算。最后，我们应用结果以在经典的等速度设置中获得等值不平等，在某些条件下，该设置在渐近上是渐近的。本文中的证明基于信息理论和最佳运输技术。
In this paper, we derive variational formulas for the asymptotic exponents (i.e., convergence rates) of the concentration and isoperimetric functions in the product Polish probability space under certain mild assumptions. These formulas are expressed in terms of relative entropies (which are from information theory) and optimal transport cost functionals (which are from optimal transport theory). Hence, our results verify an intimate connection among information theory, optimal transport, and concentration of measure or isoperimetric inequalities. In the concentration regime, the corresponding variational formula is in fact a dimension-free bound in the sense that this bound is valid for any dimension. A cardinality bound for the alphabet of the auxiliary random variable in the expression of the asymptotic isoperimetric exponent is provided, which makes the expression computable by a finite-dimensional program for the finite alphabet case. We lastly apply our results to obtain an isoperimetric inequality in the classic isoperimetric setting, which is asymptotically sharp under certain conditions. The proofs in this paper are based on information-theoretic and optimal transport techniques.
