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- W2963128082 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 $kgeq 3$; - We demonstrate a separation between testing and learning on ${0,1}^d$, for $k=omega(log d)$: testing $k$-monotonicity can be performed with $2^{O(sqrt d cdot log dcdot log{1/varepsilon})}$ queries, while learning $k$-monotone functions requires $2^{Omega(kcdot sqrt dcdot{1/varepsilon})}$ queries (Blais et al. (RANDOM 2015)). - We present a tolerant test for functions $fcolon[n]^dto {0,1}$ 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." @default.
- W2963128082 created "2019-07-30" @default.
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- W2963128082 date "2016-09-01" @default.
- W2963128082 modified "2023-09-27" @default.
- W2963128082 title "Testing $k$-Monotonicity" @default.
- W2963128082 hasPublicationYear "2016" @default.
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