The List-Decoding Size of Fourier-Sparse Boolean Functions

TL;DR

This paper establishes bounds on the number of Fourier-sparse Boolean functions close to a given function, leading to improved learning and testing algorithms with near-optimal sample complexity.

Contribution

It provides a new upper bound on the number of Fourier-sparse Boolean functions near a given function, improving learning and testing bounds.

Findings

Bound on the number of close Fourier-sparse Boolean functions

Sample complexity for learning Fourier-sparse functions is O(n·k log k)

Query complexity for testing Booleanity is nearly tight

Abstract

A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the number of $k$-Fourier-sparse Boolean functions that disagree with $f$ on at most $d$ inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of $k$-Fourier-sparse Boolean functions on $n$ variables exactly is at most $O(n \cdot k \log k)$. As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz (Chicago J. Theor. Comput. Sci., 2013).