research-article

Optimal bounds for sign-representing the intersection of two halfspaces by polynomials

Author:

Alexander A. SherstovAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 523 - 532

https://doi.org/10.1145/1806689.1806762

Published: 05 June 2010 Publication History

Abstract

The threshold degree of a function f:{0,1}ⁿ->{-1,+1} is the least degree of a real polynomial p with f=sgn p. We prove that the intersection of two halfspaces on {0,1}ⁿ has threshold degree Omega(n), which matches the trivial upper bound and completely answers a question due to Klivans (2002). The best previous lower bound was Omega(sqrt n). Our result shows that the intersection of two halfspaces on {0,1}ⁿ only admits a trivial 2^Θ(n)-time learning algorithm based on sign-representation by polynomials, unlike the advances achieved in PAC learning DNF formulas and read-once Boolean formulas. The proof introduces a new technique of independent interest, based on Fourier analysis and matrix theory.

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Cited By

Aaronson SKothari RKretschmer WThaler JAceto L(2020)Quantum lower bounds for approximate counting via laurent polynomialsProceedings of the 35th Computational Complexity Conference10.4230/LIPIcs.CCC.2020.7(1-47)Online publication date: 28-Jul-2020
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2020.7
Bun MThaler J(2015)Hardness Amplification and the Approximate Degree of Constant-Depth CircuitsAutomata, Languages, and Programming10.1007/978-3-662-47672-7_22(268-280)Online publication date: 20-Jun-2015
https://doi.org/10.1007/978-3-662-47672-7_22
Sherstov AShmoys D(2014)Breaking the minsky-papert barrier for constant-depth circuitsProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591871(223-232)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591871
Show More Cited By

Index Terms

Optimal bounds for sign-representing the intersection of two halfspaces by polynomials
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

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cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
,
Program Chair:
Leonard J. Schulman
Caltech

Copyright © 2010 ACM.

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Published: 05 June 2010

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Cited By

Aaronson SKothari RKretschmer WThaler JAceto L(2020)Quantum lower bounds for approximate counting via laurent polynomialsProceedings of the 35th Computational Complexity Conference10.4230/LIPIcs.CCC.2020.7(1-47)Online publication date: 28-Jul-2020
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2020.7
Bun MThaler J(2015)Hardness Amplification and the Approximate Degree of Constant-Depth CircuitsAutomata, Languages, and Programming10.1007/978-3-662-47672-7_22(268-280)Online publication date: 20-Jun-2015
https://doi.org/10.1007/978-3-662-47672-7_22
Sherstov AShmoys D(2014)Breaking the minsky-papert barrier for constant-depth circuitsProceedings of the forty-sixth annual ACM symposium on Theory of computing10.1145/2591796.2591871(223-232)Online publication date: 31-May-2014
https://dl.acm.org/doi/10.1145/2591796.2591871
Sherstov AKarloff HPitassi T(2012)Making polynomials robust to noiseProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214044(747-758)Online publication date: 19-May-2012
https://dl.acm.org/doi/10.1145/2213977.2214044
Vempala S(2010)A random-sampling-based algorithm for learning intersections of halfspacesJournal of the ACM10.1145/1857914.185791657:6(1-14)Online publication date: 5-Nov-2010
https://dl.acm.org/doi/10.1145/1857914.1857916
Amano K(2010)New Upper Bounds on the Average PTF Density of Boolean FunctionsAlgorithms and Computation10.1007/978-3-642-17517-6_28(304-315)Online publication date: 2010
https://doi.org/10.1007/978-3-642-17517-6_28

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