License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.54
URN: urn:nbn:de:0030-drops-94581
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9458/
O'Donnell, Ryan ;
Zhao, Yu
On Closeness to k-Wise Uniformity
Abstract
A probability distribution over {-1, 1}^n is (epsilon, k)-wise uniform if, roughly, it is epsilon-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (epsilon, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (epsilon, k)-wise uniform distribution is O(n^{k/2}epsilon)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (epsilon, k)-wise uniform distribution that is Omega(n^{k/2}epsilon)-far from any k-wise uniform distribution in total variation distance. For k=1, we get a better upper bound of O(epsilon), which is also optimal.
One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^{k}/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^{k/2}/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2.
Our results improve upon the best known bounds from [Alon et al., 2007], and have simpler proofs.
BibTeX - Entry
@InProceedings{odonnell_et_al:LIPIcs:2018:9458,
author = {Ryan O'Donnell and Yu Zhao},
title = {{On Closeness to k-Wise Uniformity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {54:1--54:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-085-9},
ISSN = {1868-8969},
year = {2018},
volume = {116},
editor = {Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9458},
URN = {urn:nbn:de:0030-drops-94581},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.54},
annote = {Keywords: k-wise independence, property testing, Fourier analysis, Boolean function}
}
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Keywords: |
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k-wise independence, property testing, Fourier analysis, Boolean function |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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13.08.2018 |
