Haviv, Ishay ;
Regev, Oded
The List-Decoding Size of Fourier-Sparse Boolean Functions
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_2^n -> 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].
BibTeX - Entry
@InProceedings{haviv_et_al:LIPIcs:2015:5060,
author = {Ishay Haviv and Oded Regev},
title = {{The List-Decoding Size of Fourier-Sparse Boolean Functions}},
booktitle = {30th Conference on Computational Complexity (CCC 2015)},
pages = {58--71},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-81-1},
ISSN = {1868-8969},
year = {2015},
volume = {33},
editor = {David Zuckerman},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5060},
URN = {urn:nbn:de:0030-drops-50600},
doi = {10.4230/LIPIcs.CCC.2015.58},
annote = {Keywords: Fourier-sparse functions, list-decoding, learning theory, property testing}
}
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Keywords: |
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Fourier-sparse functions, list-decoding, learning theory, property testing |
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Seminar: |
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30th Conference on Computational Complexity (CCC 2015)
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Issue date: |
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2015 |
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Date of publication: |
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2015 |
2015
