Fast Fourier Sparsity Testing

TL;DR

This paper introduces an efficient algorithm for estimating the $ ext{l}_2$-distance from a function to the closest $s$-sparse Fourier spectrum over $ ext{F}_2^n$, advancing sparse Fourier transform analysis in noisy settings.

Contribution

It presents the first efficient $ ext{l}_2$-distance estimation algorithm for $s$-sparse Fourier functions over $ ext{F}_2^n$, with query complexity nearly optimal.

Findings

Algorithm achieves $ ext{O}(s)$ query complexity for constant accuracy.

The method is quadratically worse than theoretical lower bounds.

Applicable to noisy Fourier spectra in high-dimensional settings.

Abstract

A function $f : \mathbb{F}_2^n \to \mathbb{R}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. Motivated by applications to fast sparse Fourier transforms over $\mathbb{F}_2^n$, we study efficient algorithms for the problem of approximating the $\ell_2$-distance from a given function to the closest $s$-sparse function. While previous works (e.g., Gopalan et al. SICOMP 2011) study the problem of distinguishing $s$-sparse functions from those that are far from $s$-sparse under Hamming distance, to the best of our knowledge no prior work has explicitly focused on the more general problem of distance estimation in the $\ell_2$ setting, which is particularly well-motivated for noisy Fourier spectra. Given the focus on efficiency, our main result is an algorithm that solves this problem with query complexity $\mathcal{O}(s)$ for constant accuracy and error parameters, which is only quadratically worse than applicable lower bounds.