Random low degree polynomials are hard to approximate

Ido Ben-Eliezer; Rani Hod; Shachar Lovett

doi:10.1007/978-3-642-03685-9_28

Random low degree polynomials are hard to approximate

Ido Ben-Eliezer^*, Rani Hod, Shachar Lovett

^*Corresponding author for this work

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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 language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings
Pages	366-377
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-642-03685-9_28
State	Published - 2009
Event	12th 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 2009 → 23 Aug 2009

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	5687 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009
Country/Territory	United States
City	Berkeley, CA
Period	21/08/09 → 23/08/09

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10.1007/978-3-642-03685-9_28

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Ben-Eliezer, I., Hod, R., & Lovett, S. (2009). Random low degree polynomials are hard to approximate. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings (pp. 366-377). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5687 LNCS). https://doi.org/10.1007/978-3-642-03685-9_28

Ben-Eliezer, Ido ; Hod, Rani ; Lovett, Shachar. / Random low degree polynomials are hard to approximate. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. pp. 366-377 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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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.",

author = "Ido Ben-Eliezer and Rani Hod and Shachar Lovett",

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doi = "10.1007/978-3-642-03685-9_28",

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Ben-Eliezer, I, Hod, R & Lovett, S 2009, Random low degree polynomials are hard to approximate. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5687 LNCS, pp. 366-377, 12th 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, 21/08/09. https://doi.org/10.1007/978-3-642-03685-9_28

Random low degree polynomials are hard to approximate. / Ben-Eliezer, Ido; Hod, Rani; Lovett, Shachar.

Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. p. 366-377 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5687 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Ben-Eliezer I, Hod R, Lovett S. Random low degree polynomials are hard to approximate. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. p. 366-377. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-03685-9_28

Random low degree polynomials are hard to approximate

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