Quadratic Goldreich--Levin Theorems

Abstract

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 [O. Goldreich and L. Levin, “A Hard-Core Predicate for All One-Way Functions," in Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 25--32] 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 is correspondingly weaker. For example, it has 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 $1/2-\e$ from a codeword. Given a function $f:\F_2^n \to \{-1,1\}$ at fractional Hamming distance $1/2-\e$ from a quadratic phase (which is a codeword of the 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 $1/2-\eta$ for $\eta = \eta(\e)$. This is an algorithmic analogue of Samorodnitsky's result [A. Samorodnitsky, “Low-Degree Tests at Large Distances,” in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 2007, pp. 506--515], 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. For this purpose, 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 $\F_2^n$.