AI 中文总结
该研究定义分析了量化极大相关性，通过率失真等技术得到其显式界，给出乘积分布下与维度无关的上界，还改进了马尔可夫链等周常数界，强化了经典结果。
AI 中文摘要
本文定义并分析了量化极大相关性，它是极大相关性概念的扩展，适用于取值于有界基数集合的函数。我们通过证明任意X和Y的量化函数之间的相关性与随机变量特定线性组合量化的MMSE（最小均方误差）失真相关，推导出量化极大相关性的上界。随后，我们利用率失真技术和反集中不等式进一步约束该MMSE，从而得到量化极大相关性的显式界。与本身通常不具备张量积性质的量化极大相关性不同，我们的均方误差界具有张量积性质，这为乘积分布下的量化极大相关性提供了与维度无关的上界。我们的结果还改进了可逆马尔可夫链及乘积链的等周常数界，强化了Alon和Milman等经典结果。
英文摘要
In this paper, we define and analyze the quantized maximal correlation, an extension of the notion of maximal correlation restricted to functions taking values in sets of bounded cardinality. We derive an upper bound on the quantized maximal correlation by showing that the correlation between any quantized functions of $X$ and $Y$ is related to the MMSE distortion in quantization of a particular linear combination of random variables. Following this, we leverage rate-distortion techniques and anti-concentration inequalities to further bound this MMSE, which results in explicit bounds on the quantized maximal correlation. Unlike the quantized maximal correlation itself, which does not generally tensorize, our bounds on the mean squared error do tensorize, resulting in a dimension-free upper bound on the quantized maximal correlation for product distributions. Our results also lead to improved bounds on the isoperimetric constants of reversible Markov chains and product chains, strengthening classical results such as those by Alon and Milman.
