Random low degree polynomials are hard to approximate

Ido Ben-Eliezer, Rani Hod, Shachar Lovett

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

4 Scopus citations

Abstract

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over double-struck F 2. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d, for all degrees d up to Θ (n). That is, a random degree d + ∈1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed-Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed-Muller codes.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings
Pages366-377
Number of pages12
DOIs
StatePublished - 2009
Externally publishedYes
Event12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 - Berkeley, CA, United States
Duration: 21 Aug 200923 Aug 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5687 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009
Country/TerritoryUnited States
CityBerkeley, CA
Period21/08/0923/08/09

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