Decompositions in theorems in classical Fourier analysis which decompose a function into large Fourier coefficients and a part that is pseudorandom
with respect to (has small correlation with) linear functions. The Goldreich-Levin theorem [GL89] can be viewed as an algorithmic version of such a decomposition as it gives an efficient algorithm for computing it. In the study of ``quadratic Fourier analysis'', higher degree analogues of such decompositions have been developed, which allow decompositions into parts with stronger properties. We develop a polynomial time algorithm for
computing a decomposition of a function into quadratic phase function and a part which is small in Gowers $U^3$ norm.
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 $1/2-\epsilon$ from a codeword. This is an algorithmic version of a result of Samorodnitsky, who gave a tester for this problem. To our knowledge, this is the first instance of a correction procedure for any class of codes, beyond the list-decoding radius.
I will describe a new constructive proof of the decomposition theorem of Gowers and Wolf [GT09], which gives a decomposition of an arbitrary bounded function, into few quadratic phases, a part that is small in Gowers $U^3$ norm and another error term which is small $\ell_1$ norm. I will then describe some components of the above correction procedure, which can be plugged into the proof of the decomposition theorem to obtain a quadratic Goldreich-Levin theorem.
Joint work with Julia Wolf.