On the structure of boolean functions with small spectral norm

Amir Shpilka, Avishay Tal, Ben Lee Volk

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

Abstract

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is ||f̂||1 = Σ α |f̂(α)|). Specifically, we prove the following results for functions f: {0, 1}n → {0,1} with ||f̂||1 = A. 1. There is a subspace V of co-dimension at most A2 such that f\V is constant. 2. f can be computed by a parity decision tree of size 2 A2n2A. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. f can be computed by a De Morgan formula of size O(2A2n2A+2) and by a De Morgan formula of depth O(A2 + log(n) · A). 4. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth A2 log s. 5. For every ε > 0 there is a parity decision tree of depth O(A2 + log(1/ε)) and size 2 O(A2) · min{1/ε2, O(log(1/ε)) 2A} that e-approximates f. Furthermore, this tree can be learned, with probability 1 - δ, using poly(n, exp(A2), 1/ε, log(1/δ)) membership queries. All the results above (except 3) also hold (with a slight change in parameters) for functions f: ℤp n → {0,1}.

Original languageEnglish
Title of host publicationITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science
PublisherAssociation for Computing Machinery
Pages37-47
Number of pages11
ISBN (Print)9781450322430
DOIs
StatePublished - 2014
Externally publishedYes
Event2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014 - Princeton, NJ, United States
Duration: 12 Jan 201414 Jan 2014

Publication series

NameITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science

Conference

Conference2014 5th Conference on Innovations in Theoretical Computer Science, ITCS 2014
Country/TerritoryUnited States
CityPrinceton, NJ
Period12/01/1414/01/14

Keywords

  • Analysis of Boolean Functions
  • Decision Trees
  • Spectral Norm

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