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Fourier Bounds and Pseudorandom Generators for Product Tests

Author Chin Ho Lee

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • bounded independence plus noise
  • Fourier spectrum
  • product test
  • pseudorandom generators

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Abstract

We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest.
As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths. 
We also extend our results to functions f_i whose range is [-1,1].

Cite As Get BibTex

Chin Ho Lee. Fourier Bounds and Pseudorandom Generators for Product Tests. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 7:1-7:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CCC.2019.7

Author Details

Chin Ho Lee
  • Northeastern University, Boston, USA

Funding

Supported by NSF CCF award 1813930. Work done in part while visiting Amnon Ta-Shma at Tel Aviv University, with support from the Blavatnik family fund and ISF grant no. 952/18.

Acknowledgements

I thank Salil Vadhan for asking about the Fourier spectrum of product tests. I also thank Andrej Bogdanov, Gil Cohen, Amnon Ta-Shma, Avishay Tal and Emanuele Viola for very helpful conversations. I am grateful to Emanuele Viola for his invaluable comments on the write-up, and Dean Doron, Pooya Hatami and William Hoza for pointing out a simplification and improvement of Theorem 8. Finally, I thank the anonymous reviewers for their useful comments.

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