二次Goldreich-Levin定理
Quadratic Goldreich-Levin Theorems
摘要：
经典傅立叶分析中的分解定理使我们能够用几个具有大傅立叶系数的线性相位加上相对于线性相位是伪随机的部分来表示有界函数。Goldreich-Levin算法可以被看作是这种分解的算法模拟，因为它给出了一种有效地找到与大傅立叶系数相关联的线性相位的方法。在"二次傅立叶分析"的研究中，已经开发了这种分解的更高程度的类似物，其中伪随机性更强，但结构部分相应地更弱。例如，先前已经表明，可以将有界函数表示为几个二次相位加上在$U^3$范数中较小的部分的总和，该范数由Gowers定义，用于计数长度为4的算术级数。我们给出了一个计算这种分解的多项式时间算法。该算法的一个关键部分是一个局部自校正程序的里德-穆勒码的顺序2（超过$\F_2^n$）的功能在距离$1/2-ε$从码字。给定一个函数$f:\F_2^n \to \{-1,1\}$，它与二次相位（二阶Reed-Muller码的码字）的汉明距离为$1/2-ε$的分数倍，本文给出了一个在$n$中按时间多项式运行的算法，该算法为$η= η(ε)$找到距离至多为$1/2-η$的码字.这是一个算法模拟Samorodnitsky的结果，这给了一个测试上述问题。据我们所知，它代表了任何类别的代码的纠正过程的第一个实例，超出了列表解码半径。在这个过程中，我们给算法版本的结果添加剂组合在Samorodnitsky的证明和一个改进版本的逆定理的Gowers $U^3$规范$\F_2^n$。
英文摘要：
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 $U^3$ 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 $\F_2^n$) for a function at distance $1/2-ε$ from a codeword. Given a function $f:\F_2^n \to \{-1,1\}$ at fractional Hamming distance $1/2-ε$ 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 $1/2-η$ for $η= η(ε)$. 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 $U^3$ norm over $\F_2^n$.
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