Testing Surface Area

Kothari, Pravesh K.; Nayyeri, Amir; O’Donnell, Ryan; Wu, Chengyou

doi:10.1137/1.9781611973402.89

Cited by 16 publications

(31 citation statements)

References 15 publications

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“…Namely, the algorithm rejects functions that are -far from every 4k-junta rather than -far from every k-junta. Similar relaxations have been considered both in the standard testing model (e.g., [32,26,28]) and in the tolerant testing model [33]. Although one may hope for a stronger statement (distinguishing functions close to k-junta from those far from k-junta), we note that for most practical purposes the relaxation above is more than sufficient.…”

Section: Our Resultsmentioning

confidence: 85%

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Blais

Canonne

Eden

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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The function f : {−1, 1} n → {−1, 1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is -close to some k-junta and reject any function that is -far from every k -junta for some = O( ) and k = O(k).Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ . This result is obtained via a new polynomialtime approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function.Our second result considers the case where k = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that given ρ ∈ (0, 1/2) accepts any function that is ρ 8 -close to some k-junta and rejects any function that is -far from every k-junta. The query complexity of the algorithm is OFinally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.

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Section: Our Resultsmentioning

confidence: 85%

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Blais

Canonne

Eden

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…A well-known link between Gaussian surface area and Hermite expansions then implies that the rounded, smoothed function is almost a low-degree PTF. This argument uses the co-area formula, gradient bounds and is inspired by ideas from [KNOW14,Nee14].…”

Section: Introductionmentioning

confidence: 99%

Non interactive simulation of correlated distributions is decidable

De¹,

Mossel

Neeman

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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A basic problem in information theory is the following: Let P = (X, Y) be an arbitrary distribution where the marginals X and Y are (potentially) correlated. Let Alice and Bob be two players where Alice gets samples {x i } i≥1 and Bob gets samples {y i } i≥1 and for all i, (x i , y i ) ∼ P. What joint distributions Q can be simulated by Alice and Bob without any interaction?Classical works in information theory by Gács-Körner and Wyner answer this question when at least one of P or Q is the distribution Eq (Eq is defined as uniform over the points (0, 0) and (1, 1)). However, other than this special case, the answer to this question is understood in very few cases. Recently, Ghazi, Kamath and Sudan showed that this problem is decidable for Q supported on {0, 1} × {0, 1}. We extend their result to Q supported on any finite alphabet. Moreover, we show that If Q can be simulated, our algorithm also provides a (non-interactive) simulation protocol.We rely on recent results in Gaussian geometry (by the authors) as well as a new smoothing argument inspired by the method of boosting from learning theory and potential function arguments from complexity theory and additive combinatorics.

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“…Testing with two-sided error, however, does not require a dependence on k. In fact the problem has been well-studied in the machine learning literature in the context of testing/learning "union of intervals" [35,5], and in testing geometric properties, in the context of testing surface area [38,42], 2 resulting in an O(1/ε 7/2 )-query algorithm. Namely, the starting point of [5] (later improved by [38]) is a "Buffon's Needle"-type argument. There, the crucial quantity to analyze is the noise sensitivity of the function, that is the probability that a randomly chosen pair of nearby points cross a "boundary," i. e., have different values.…”

Section: Testing K-monotonicity On the Hypercube {0 1} Dmentioning

confidence: 99%

Untitled

Canonne¹,

Grigorescu²,

Guo³

et al. 2019

Theory of Comput.

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A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, kmonotone 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. 1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0, 1} d , for k ≥ 3;

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Testing Surface Area

Cited by 16 publications

References 15 publications

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Non interactive simulation of correlated distributions is decidable

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