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Kevin Matulef

Bio: Kevin Matulef is an academic researcher from Tsinghua University. The author has contributed to research in topics: Property testing & Boolean function. The author has an hindex of 10, co-authored 16 publications receiving 732 citations. Previous affiliations of Kevin Matulef include Aarhus University & Massachusetts Institute of Technology.

Papers
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Journal ArticleDOI
08 Jun 2011
TL;DR: In this article, a technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity, was developed, which is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds.
Abstract: We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Omega(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

150 citations

Proceedings ArticleDOI
11 Jun 2007
TL;DR: This work gives evidence that the time complexity of testing (ε,k)-wise independence is likely to be poly(n, 1/ε,1/δ) for k=Θ(log n), since this would disprove a plausible conjecture concerning the hardness offinding hidden cliques in random graphs.
Abstract: In this work, we consider the problems of testing whether adistribution over (0,1n) is k-wise (resp. (e,k)-wise) independentusing samples drawn from that distribution.For the problem of distinguishing k-wise independent distributions from those that are δ-far from k-wise independence in statistical distance, we upper bound the number ofrequired samples by O(nk/δ2) and lower bound it by Ω(nk-1/2/δ) (these bounds hold for constantk, and essentially the same bounds hold for general k). Toachieve these bounds, we use Fourier analysis to relate adistribution's distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a setof variables. The relationships we derive are tighter than previouslyknown, and may be of independent interest.To distinguish (e,k)-wise independent distributions from thosethat are δ-far from (e,k)-wise independence in statistical distance, we upper bound thenumber of required samples by O(k log n / δ2e2) and lower bound it by Ω(√ k log n / 2k(e+δ)√ log 1/2k(e+δ)). Although these bounds are anexponential improvement (in terms of n and k) over thecorresponding bounds for testing k-wise independence, we give evidence thatthe time complexity of testing (e,k)-wise independence isunlikely to be poly(n,1/e,1/δ) for k=Θ(log n),since this would disprove a plausible conjecture concerning the hardness offinding hidden cliques in random graphs. Under the conjecture, ourresult implies that for, say, k = log n and e = 1 / n0.99,there is a set of (e,k)-wise independent distributions, and a set of distributions at distance δ=1/n0.51 from (e,k)-wiseindependence, which are indistinguishable by polynomial time algorithms.

137 citations

Journal Article
TL;DR: A new technique for proving lower bounds in property testing is developed, by showing a strong connection between testing and communication complexity, and significantly strengthens the best known bounds.
Abstract: We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Omega(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

107 citations

Proceedings ArticleDOI
04 Jan 2009
TL;DR: In this paper, the authors considered the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w · x - θ).
Abstract: This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w · x - θ). We consider halfspaces over the continuous domain Rn (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {−1, 1}n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are e-far from any halfspace using only poly(1/e) queries, independent of the dimension n.Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: the first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f's degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube {−1, 1}n (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over {−1, 1}n poses significant additional complications and requires other ingredients. These include "cross-consistency" versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [FKR+02].

89 citations

Journal ArticleDOI
TL;DR: This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w · x - θ) by giving an algorithm that distinguishes halfspaces from functions that are e-far from any halfspace using only poly(1/e) queries, independent of the dimension n.
Abstract: This paper addresses the problem of testing whether a Boolean-valued function $f$ is a halfspace, i.e., a function of the form $f(x)=\mathrm{sgn}(w\cdot x-\theta)$. We consider halfspaces over the continuous domain $\mathbf{R}^n$ (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube $\{-1,1\}^n$ (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are $\epsilon$-far from any halfspace using only $\mathrm{poly}(\frac{1}{\epsilon})$ queries, independent of the dimension $n$. Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: The first gives an exact relationship between the expected value of a halfspace $f$ and the sum of the squares of $f$'s degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube $\{-1,1\}^n$ (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over $\{-1,1\}^n$ poses significant additional complications and requires other ingredients. These include “cross-consistency” versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [E. Fischer, G. Kindler, D. Ron, S. Safra, and A. Samorodnitsky, Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, 2002, pp. 103-112].

86 citations


Cited by
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Book
02 Jan 1991

1,377 citations

Journal Article
TL;DR: In this paper, the authors consider the question of determining whether a function f has property P or is e-far from any function with property P. In some cases, it is also allowed to query f on instances of its choice.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

870 citations

Book
05 Jun 2014
TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

867 citations

MonographDOI
01 Jan 2014

575 citations

Book ChapterDOI
01 Jan 1996

378 citations