### Abstract

A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most knonzero Fourier coefficients. For a function f: F^{n} _{2} → ℝ and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n·klog k). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [2013].

Original language | English (US) |
---|---|

Article number | 10 |

Journal | ACM Transactions on Computation Theory |

Volume | 8 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2016 |

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### Keywords

- Fourier-sparse Boolean functions
- Learning theory
- Property testing

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Theoretical Computer Science

### Cite this

**The list-decoding size of fourier-sparse Boolean functions.** / Haviv, Ishay; Regev, Oded.

Research output: Contribution to journal › Article

*ACM Transactions on Computation Theory*, vol. 8, no. 3, 10. https://doi.org/10.1145/2898439

}

TY - JOUR

T1 - The list-decoding size of fourier-sparse Boolean functions

AU - Haviv, Ishay

AU - Regev, Oded

PY - 2016/5/1

Y1 - 2016/5/1

N2 - A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most knonzero Fourier coefficients. For a function f: Fn 2 → ℝ and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n·klog k). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [2013].

AB - A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most knonzero Fourier coefficients. For a function f: Fn 2 → ℝ and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n·klog k). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [2013].

KW - Fourier-sparse Boolean functions

KW - Learning theory

KW - Property testing

UR - http://www.scopus.com/inward/record.url?scp=84969924819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969924819&partnerID=8YFLogxK

U2 - 10.1145/2898439

DO - 10.1145/2898439

M3 - Article

VL - 8

JO - ACM Transactions on Computation Theory

JF - ACM Transactions on Computation Theory

SN - 1942-3454

IS - 3

M1 - 10

ER -