Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Let [Formula: see text] , [Formula: see text] , be the noise operator acting on functions on the boolean cube [Formula: see text]. Let f be a distribution on [Formula: see text] and let [Formula: see text]. We prove tight Mrs. Gerber-type results for the second Rényi entropy of [Formula: see text] w...

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Autores principales: Levhari, Niv, Samorodnitsky, Alex
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2022
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9601360/
https://www.ncbi.nlm.nih.gov/pubmed/37420395
http://dx.doi.org/10.3390/e24101376
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author Levhari, Niv
Samorodnitsky, Alex
author_facet Levhari, Niv
Samorodnitsky, Alex
author_sort Levhari, Niv
collection PubMed
description Let [Formula: see text] , [Formula: see text] , be the noise operator acting on functions on the boolean cube [Formula: see text]. Let f be a distribution on [Formula: see text] and let [Formula: see text]. We prove tight Mrs. Gerber-type results for the second Rényi entropy of [Formula: see text] which take into account the value of the [Formula: see text] Rényi entropy of f. For a general function f on [Formula: see text] we prove tight hypercontractive inequalities for the [Formula: see text] norm of [Formula: see text] which take into account the ratio between [Formula: see text] and [Formula: see text] norms of f.
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spelling pubmed-96013602022-10-27 Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy Levhari, Niv Samorodnitsky, Alex Entropy (Basel) Article Let [Formula: see text] , [Formula: see text] , be the noise operator acting on functions on the boolean cube [Formula: see text]. Let f be a distribution on [Formula: see text] and let [Formula: see text]. We prove tight Mrs. Gerber-type results for the second Rényi entropy of [Formula: see text] which take into account the value of the [Formula: see text] Rényi entropy of f. For a general function f on [Formula: see text] we prove tight hypercontractive inequalities for the [Formula: see text] norm of [Formula: see text] which take into account the ratio between [Formula: see text] and [Formula: see text] norms of f. MDPI 2022-09-27 /pmc/articles/PMC9601360/ /pubmed/37420395 http://dx.doi.org/10.3390/e24101376 Text en © 2022 by the authors. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Levhari, Niv
Samorodnitsky, Alex
Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy
title Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy
title_full Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy
title_fullStr Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy
title_full_unstemmed Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy
title_short Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy
title_sort hypercontractive inequalities for the second norm of highly concentrated functions, and mrs. gerber’s-type inequalities for the second rényi entropy
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9601360/
https://www.ncbi.nlm.nih.gov/pubmed/37420395
http://dx.doi.org/10.3390/e24101376
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