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Testing $k$-Monotonicity | | Clément L. Canonne
; Elena Grigorescu
; Siyao Guo
; Akash Kumar
; Karl Wimmer
; | | Date: |
1 Sep 2016 | | 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 $fcolon[n]^d o {0,1}$ 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. | | Source: | arXiv, 1609.0265 | | Services: | Forum | Review | PDF | Favorites | |
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