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.ITCS.2017.29
URN: urn:nbn:de:0030-drops-81583
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8158/
Canonne, Clément L. ;
Grigorescu, Elena ;
Guo, Siyao ;
Kumar, Akash ;
Wimmer, Karl
Testing k-Monotonicity
Abstract
A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions.
Motivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone.
Our results include the following:
1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0,1}^d, for k >= 3;
2. We demonstrate a separation between testing and learning on {0,1}^d, for k=\omega(\log d): testing k-monotonicity can be performed with 2^{O(\sqrt d . \log d . \log{1/\eps})} queries, while learning k-monotone functions requires 2^{\Omega(k . \sqrt d .{1/\eps})} queries (Blais et al. (RANDOM 2015)).
3. We present a tolerant test for functions f\colon[n]^d\to \{0,1\}$with complexity independent of n, which makes progress on a problem left open by Berman et al. (STOC 2014).
Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.
Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.
BibTeX - Entry
@InProceedings{canonne_et_al:LIPIcs:2017:8158,
author = {Cl{\'e}ment L. Canonne and Elena Grigorescu and Siyao Guo and Akash Kumar and Karl Wimmer},
title = {{Testing k-Monotonicity}},
booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages = {29:1--29:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-029-3},
ISSN = {1868-8969},
year = {2017},
volume = {67},
editor = {Christos H. Papadimitriou},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8158},
URN = {urn:nbn:de:0030-drops-81583},
doi = {10.4230/LIPIcs.ITCS.2017.29},
annote = {Keywords: Boolean Functions, Learning, Monotonicity, Property Testing}
}
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Keywords: |
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Boolean Functions, Learning, Monotonicity, Property Testing |
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Collection: |
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8th Innovations in Theoretical Computer Science Conference (ITCS 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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28.11.2017 |
