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Free, publiclyaccessible full text available December 1, 2023

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 to monotone (namely, at a distance $O(1/\log^2 n)$). In addition to that, we consider a restricted error model that aims at capturing “missing data” corruptions. In this model, we show that distributions that are close to monotone have sampling correctors that are significantly more efficient than achievable by the learning approach. We consider the question of whether an additional source of independent random bits is required by sampling correctors to implement the correction process. We show that for correcting closetouniform distributions and closetomonotone distributions, no additional source of random bits is required, as the samples from the input source itself can be used to produce this randomness.more » « less

We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution D over [n] and a property P, the goal is to distinguish between D ∈ P and ℓ1(D, P) > ε. We develop a general algorithm for this question, which applies to a large range of “shapeconstrained” properties, including monotone, logconcave, tmodal, piecewisepolynomial, and Poisson Binomial distributions. Moreover, for all cases considered, our algorithm has nearoptimal sample complexity with regard to the domain size and is computationally efficient. For most of these classes, we provide the first nontrivial tester in the literature. In addition, we also describe a generic method to prove lower bounds for this problem, and use it to show our upper bounds are nearly tight. Finally, we extend some of our techniques to tolerant testing, deriving nearly–tight upper and lower bounds for the corresponding questions.more » « less

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.more » « less