### Abstract

A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most k nonzero 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 · k log 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 (Chicago J. Theor. Comput. Sci., 2013).

Original language | English (US) |
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Title of host publication | 30th Conference on Computational Complexity, CCC 2015 |

Editors | David Zuckerman |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 58-71 |

Number of pages | 14 |

ISBN (Electronic) | 9783939897811 |

DOIs | |

State | Published - Jun 1 2015 |

Event | 30th Conference on Computational Complexity, CCC 2015 - Portland, United States Duration: Jun 17 2015 → Jun 19 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 33 |

ISSN (Print) | 1868-8969 |

### Other

Other | 30th Conference on Computational Complexity, CCC 2015 |
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Country | United States |

City | Portland |

Period | 6/17/15 → 6/19/15 |

### Fingerprint

### Keywords

- Fourier-sparse functions
- Learning theory
- List-decoding
- Property testing

### ASJC Scopus subject areas

- Software

### Cite this

*30th Conference on Computational Complexity, CCC 2015*(pp. 58-71). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 33). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.CCC.2015.58