Towards a proof of the fourier-entropy conjecture?

Esty Kelman, Guy Kindler, Noam Lifshitz, Dor Minzer, Muli Safra

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is o(log n). However, both results become useless when the total influence of the function is omega(log n). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of S {n}. In this paper, we build and improve on the techniques of the Bourgain-Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1)A quantitative improvement of the Bourgain-Kalai result regarding the total influence of functions that are transitively symmetric. 2)A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on 2{O(I[f] log I[f])} characters, improving an earlier result of Friedgut. Removing the log I[f] factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time 2{O(K log K)} using membership queries. Thus, the class of functions with total influence O(log n log log n) is agnostically learnable in text{poly}(n) time.

Original languageEnglish
Title of host publicationProceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020
PublisherIEEE Computer Society
Pages247-258
Number of pages12
ISBN (Electronic)9781728196213
DOIs
StatePublished - Nov 2020
Event61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States
Duration: 16 Nov 202019 Nov 2020

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2020-November
ISSN (Print)0272-5428

Conference

Conference61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020
Country/TerritoryUnited States
CityVirtual, Durham
Period16/11/2019/11/20

Funding

FundersFunder number
National Science FoundationDMS-1638352
Horizon 2020 Framework Programme
European Research Council
Horizon 2020835152

    Keywords

    • Fourier analysis
    • Fourier-Entropy Conjecture
    • Learning sparse functions

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