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The List-Decoding Size of Fourier-Sparse Boolean Functions

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Published:11 May 2016Publication History
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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: F2n → 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].

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 8, Issue 3
      May 2016
      105 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/2936308
      Issue’s Table of Contents

      Copyright © 2016 ACM

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 11 May 2016
      • Revised: 1 December 2015
      • Accepted: 1 December 2015
      • Received: 1 July 2015
      Published in toct Volume 8, Issue 3

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