Problem
Research questions and friction points this paper is trying to address.
Quantized Maximal Correlation
Cardinality Constraints
Upper Bound
Product Distributions
Maximal correlation under cardinality constraints
🤖 AI Summary
This study addresses the characterization of quantized maximal correlation under cardinality constraints by integrating rate-distortion theory, anti-concentration inequalities, and MMSE analysis into a unified theoretical framework. By establishing an intrinsic connection between quantized maximal correlation and MMSE, this work proposes dimension-free upper bounds that are amenable to tensorization. Key contributions include the derivation of explicit bounds for quantized maximal correlation, the attainment of dimension-free upper bounds under product distributions, and significant improvements to classical results on isoperimetric constants for reversible Markov chains. Collectively, these findings provide novel analytical tools and theoretical foundations for the intersection of high-dimensional probability analysis and information theory.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantized Maximal Correlation
Cardinality Constraints
Rate-Distortion Theory
Tensorization
Isoperimetric Constants
🔎 Similar Papers
No similar papers found.
