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 ofmore »
Sampling Correctors
In many situations, sample data is obtained from a noisy or imperfect source. In order to address such corruptions, this paper introduces the concept of a sampling corrector. Such algorithms use structure that the distribution is purported to have, in order to allow one to make “onthefly” corrections to samples drawn from probability distributions. These algorithms then act as filters between the noisy data and the end user. We show connections between sampling correctors, distribution learning algorithms, and distribution property testing algorithms. We show that these connections can be utilized to expand the applicability of known distribution learning and property testing algorithms as well as to achieve improved algorithms for those tasks. As a first step, we show how to design sampling correctors using proper learning algorithms. We then focus on the question of whether algorithms for sampling correctors can be more efficient in terms of sample complexity than learning algorithms for the analogous families of distributions. When correcting monotonicity, we show that this is indeed the case when also granted query access to the cumulative distribution function. We also obtain sampling correctors for monotonicity even without this stronger type of access, provided that the distribution be originally very close more »
 Award ID(s):
 1650733
 Publication Date:
 NSFPAR ID:
 10078630
 Journal Name:
 SIAM journal on computing
 Volume:
 47
 Issue:
 4
 Page Range or eLocationID:
 13731423
 ISSN:
 00975397
 Sponsoring Org:
 National Science Foundation
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