Testing $k$Monotonicity
Abstract
A Boolean $k$monotone function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. 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:  We demonstrate a separation between testing $k$monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$;  We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\varepsilon})}$ queries, while learning $k$monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\varepsilon})}$ queries (Blais et al. (RANDOM 2015)).  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 testingbylearning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.00265
 Bibcode:
 2016arXiv160900265C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics;
 Computer Science  Machine Learning