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标题：Quadratic Goldreich-Levin Theorems
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作者：Madhur Tulsiani ; Julia Wolf
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2011
卷号：2011
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin algorithm can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients. In the study of ``quadratic Fourier analysis'', higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part correspondingly weaker. For example, it has previously been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the U3 norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition. A key part of the algorithm is a local self-correction procedure for Reed-Muller codes of order 2 (over Fn2) for a function at distance 12− from a codeword. Given a function f:Fn2−11 at fractional Hamming distance 12− from a quadratic phase (which is a codeword of Reed-Muller code of order 2), we give an algorithm that runs in time polynomial in n and finds a codeword at distance at most 12− for =() . This is an algorithmic analogue of Samorodnitsky's result, which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the list-decoding radius. In the process, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers U3 norm over Fn2.