The problem of testing monotonicity for Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of KhotMinzerSafra (FOCS 2015) gave a nonadaptive, onesided tester making $\otilde(\eps^{2}\sqrt{d})$ queries. Up to polylog $d$ and $\eps$ factors, this bound matches the $\widetilde{\Omega}(\sqrt{d})$query nonadaptive lower bound (ChenDeServedioTan (STOC 2015), ChenWaingartenXie (STOC 2017)). For any $n > 2$, the optimal nonadaptive complexity was unknown. A previous result of the authors achieves a $\otilde(d^{5/6})$query upper bound (SODA 2020), quite far from the $\sqrt{d}$ bound for the hypercube. In this paper, we resolve the nonadaptive complexity of monotonicity testing for all constant $n$, up to $\poly(\eps^{1}\log d)$ factors. Specifically, we give a nonadaptive, onesided monotonicity tester making $\otilde(\eps^{2}n\sqrt{d})$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]^d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.
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Testing kMonotonicity
A Boolean {\em $k$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 {\em monotone} functions, which are the {\em $1$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:
\begin{enumerate}
\item We demonstrate a separation between testing $k$monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$;
\item We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$ queries, while learning $k$monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$ queries (Blais et al. (RANDOM 2015)).
\item 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).
\end{enumerate}
Our techniques exploit the testingbylearning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques.
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 Award ID(s):
 1649515
 NSFPAR ID:
 10033552
 Date Published:
 Journal Name:
 ITCS
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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