Cubic Goldreich-Levin

In this paper, we give a cubic Goldreich-Levin algorithm which makes polynomially-many queries to a function (Equation presented) and produces a decomposition of f as a sum of cubic phases and a small error term. This is a natural higher-order generalization of the classical Goldreich-Levin algorithm. The classical (linear) Goldreich-Levin algorithm has wide-ranging applications in learning theory, coding theory and the construction of pseudorandom generators in cryptography, as well as being closely related to Fourier analysis. Higher-order Goldreich-Levin algorithms on the other hand involve central problems in higher-order Fourier analysis, namely the inverse theory of the Gowers U^k norms, which are well-studied in additive combinatorics. The only known result in this direction prior to this work is the quadratic Goldreich-Levin theorem, proved by Tulsiani and Wolf in 2011. The main step of their result involves an algorithmic version of the U³ inverse theorem. More complications appear in the inverse theory of the U⁴ and higher norms. Our cubic Goldreich-Levin algorithm is based on algorithmizing recent work by Gowers and Milićević who proved new quantitative bounds for the U⁴ inverse theorem. Our cubic Goldreich-Levin algorithm is constructed from two main tools: an algorithmic U⁴ inverse theorem and an arithmetic decomposition result in the style of the Frieze-Kannan graph regularity lemma. As one application of our main theorem we solve the problem of self-correction for cubic Reed-Muller codes beyond the list decoding radius. Additionally we give a purely combinatorial result: an improvement of the quantitative bounds on the U⁴ inverse theorem.