When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.29
URN: urn:nbn:de:0030-drops-81583
URL: https://drops.dagstuhl.de/opus/volltexte/2017/8158/
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### Testing k-Monotonicity

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### 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},