{"title":"Rényi Resolvability, Noise Stability, and Anti-Contractivity","authors":"Lei Yu","doi":"10.1109/TIT.2025.3578324","DOIUrl":null,"url":null,"abstract":"This paper investigates three closely related topics—Rényi resolvability, noise stability, and anti-contractivity. The Rényi resolvability problem refers to approximating a target output distribution of a given channel in the Rényi divergence when the input is set to a function of a given uniform random variable. This problem for the Rényi parameter in (<inline-formula> <tex-math>$0,2]\\cup \\{\\infty \\}$ </tex-math></inline-formula> was first studied by the present author and Tan in 2019. In the present paper, we provide a complete solution to this problem for the Rényi parameter in the entire range <inline-formula> <tex-math>$\\mathbb {R}\\cup \\{\\pm \\infty \\}$ </tex-math></inline-formula>. We then connect the Rényi resolvability problem to the noise stability problem, by observing that maximizing or minimizing the <italic>q</i>-stability of a set is equivalent to a variant of the Rényi resolvability problem. By such a connection, we provide sharp dimension-free bounds on the <italic>q</i>-stability. We lastly relate the noise stability problem to the anti-contractivity of a Markov operator (i.e., conditional expectation operator), where the terminology “anti-contractivity” introduced by us refers to as the opposite property of the well-known contractivity/hyercontractivity. We derive sharp dimension-free anti-contractivity inequalities. All of the results in this paper are evaluated for binary distributions. Our proofs in this paper are mainly based on the method of types, especially strengthened versions of packing-covering lemmas.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 8","pages":"5836-5867"},"PeriodicalIF":2.9000,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE Transactions on Information Theory","FirstCategoryId":"94","ListUrlMain":"https://ieeexplore.ieee.org/document/11029606/","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper investigates three closely related topics—Rényi resolvability, noise stability, and anti-contractivity. The Rényi resolvability problem refers to approximating a target output distribution of a given channel in the Rényi divergence when the input is set to a function of a given uniform random variable. This problem for the Rényi parameter in ($0,2]\cup \{\infty \}$ was first studied by the present author and Tan in 2019. In the present paper, we provide a complete solution to this problem for the Rényi parameter in the entire range $\mathbb {R}\cup \{\pm \infty \}$ . We then connect the Rényi resolvability problem to the noise stability problem, by observing that maximizing or minimizing the q-stability of a set is equivalent to a variant of the Rényi resolvability problem. By such a connection, we provide sharp dimension-free bounds on the q-stability. We lastly relate the noise stability problem to the anti-contractivity of a Markov operator (i.e., conditional expectation operator), where the terminology “anti-contractivity” introduced by us refers to as the opposite property of the well-known contractivity/hyercontractivity. We derive sharp dimension-free anti-contractivity inequalities. All of the results in this paper are evaluated for binary distributions. Our proofs in this paper are mainly based on the method of types, especially strengthened versions of packing-covering lemmas.
期刊介绍:
The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.