Quadratic Goldreich--Levin Theorems

Tulsiani, Madhur; Wolf, Julia

doi:10.1137/12086827x

Cited by 5 publications

(1 citation statement)

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“…The only such result currently known is by Tulsiani and Wolf [TW11] who give a decoding algorithm that works for k = 2 over F 2 . For larger k, the question is open.…”

Section: Definition 53 (Gowers Norm)mentioning

confidence: 99%

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Bhattacharyya

2013

SIGACT News

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An affine-invariant property over a finite field is a property of functions over F n p that is closed under all affine transformations of the domain. This class of properties includes such well-known beasts as low-degree polynomials, polynomials that nontrivially factor, and functions of low spectral norm. The last few years has seen rapid progress in characterizing the affine-invariant properties which are testable with a constant number of queries. We survey the current state of this project. What Is Property Testing?A scientific experiment takes as input an unknown object 3 , and it aims to determine whether or not a certain statement about the object holds true. Often, it's not feasible to decide whether the statement is exactly true or not, and the experimenter is satisfied with knowing whether the statement is "approximately true" for the object. The experiment consists of making certain kinds of measurements on the object, and one usually wants to minimize the number of measurements needed to reach a conclusion. Property testing is an algorithmic formalization of this basic scientific endeavor.Let O be a set of objects, and let P be the subset of O that satisfies a certain desirable property. Given an unknown object o ∈ O, we wish to know whether o is "close to being a member of P" by making a small number of "measurements" on o. To make this precise, introduce a distance function dist between pairs of objects in O and a query model that specifies what the possible queries (measurements) into o are and how each query reveals information about o. Then, for a given parameter ε > 0 that parameterizes the level of approximation and a positive integer q, an (ε, q)-tester for P is an algorithm 4 A that makes q queries into an unknown input o, outputs yes if o is in P and no if dist(o, o ) > ε for every o in P. That is, if the algorithm A outputs yes, then there is a guarantee that dist(o, o ) ε for some o in P, and hence, P approximately holds true for o. 1 . arnabb@csa.iisc.ernet.in.3 Well, not completely unknown. The experimenter usually already has some partial information about the object. 4 The algorithm is randomized, and the output guarantees hold with constant probability over the randomness of the algorithm. We give a precise definition later.

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“…The only such result currently known is by Tulsiani and Wolf [TW11] who give a decoding algorithm that works for k = 2 over F 2 . For larger k, the question is open.…”

Section: Definition 53 (Gowers Norm)mentioning

confidence: 99%

Guest column

Bhattacharyya

2013

SIGACT News

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Some applications of relative entropy in additive combinatorics

Wolf¹

2017

Surveys in Combinatorics 2017

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Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

Bhattacharyya

Gopi

2017

ACM Trans. Comput. Theory

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Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well-suited to admit local correctors and testers.In this work, we give lower bounds on the length of locally correctable and locally testable affineinvariant codes with constant query complexity. We show that if a code C ⊂ Σ K n is an r-query locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp(O K,r,|Σ| (n r−1 )). Also, we show that if C ⊂ Σ K n is an r-query locally testable code (LTC), then the number of codewords in C is at most exp(O K,r,|Σ| (n r−2 )). The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS '13) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM '11) assumed linearity to derive similar results.Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, upto a small error in the Gowers norm.

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Quadratic Goldreich--Levin Theorems

Cited by 5 publications

References 29 publications

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Some applications of relative entropy in additive combinatorics

Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

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