A near-optimal Quadratic Goldreich-Levin algorithm

TL;DR

This paper presents a near-optimal quadratic Goldreich-Levin algorithm that efficiently finds quadratic phase functions correlating with a given Boolean function, with applications to coding theory and Gowers norms.

Contribution

It introduces a quadratic Goldreich-Levin algorithm that is close to optimal in query complexity and runtime, improving previous methods and avoiding reliance on the polynomial Freiman-Ruzsa conjecture.

Findings

Algorithm runs in $O_\u03b5(n^3)$ time with $O_\u03b5(n^2 \u2212  ext{log} n)$ queries.

Provides a near-optimal self-corrector for quadratic Reed-Muller codes.

Establishes an algorithmic inverse theorem for the order-3 Gowers uniformity norm.

Abstract

In this paper, we give a quadratic Goldreich-Levin algorithm that is close to optimal in the following ways. Given a bounded function $f$ on the Boolean hypercube $\mathbb{F}_2^n$ and any $\varepsilon>0$, the algorithm returns a quadratic polynomial $q: \mathbb{F}_2^n \to \mathbb{F}_2$ so that the correlation of $f$ with the function $(-1)^q$ is within an additive $\varepsilon$ of the maximum possible correlation with a quadratic phase function. The algorithm runs in $O_\varepsilon(n^3)$ time and makes $O_\varepsilon(n^2\log n)$ queries to $f$, which matches the information-theoretic lower bound of $\Omega(n^2)$ queries up to a logarithmic factor. As a result, we obtain a number of corollaries: - A near-optimal self-corrector of quadratic Reed-Muller codes, which makes $O_\varepsilon(n^2\log n)$ queries to a Boolean function $f$ and returns a quadratic polynomial $q$ whose relative Hamming distance to $f$ is within $\varepsilon$ of the minimum distance. - An algorithmic polynomial inverse theorem for the order-3 Gowers uniformity norm. - An algorithm that makes a polynomial number of queries to a bounded function $f$ and decomposes $f$ as a sum of poly$(1/\varepsilon)$ quadratic phase functions and error terms of order $\varepsilon$. Our algorithm is obtained using ideas from recent work on quantum learning theory. Its construction deviates from previous approaches based on algorithmic proofs of the inverse theorem for the order-3 uniformity norm (and in particular does not rely on the recent resolution of the polynomial Fre\u{\i}man-Ruzsa conjecture).