On the Structure of Boolean Functions with Small Spectral Norm

TL;DR

This paper investigates Boolean functions with small spectral norm, establishing structural properties, decision tree representations, and approximation capabilities, with implications for learning theory and Fourier analysis.

Contribution

It provides new bounds and constructions for Boolean functions with small spectral norm, including structural subspaces, decision trees, and approximation algorithms.

Findings

Existence of a low-co-dimension subspace where the function is constant

Efficient parity decision tree representations

Approximation of functions with small spectral norm using shallow trees

Abstract

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n \to \{0,1\}$ with $\|\hat{f}\|_1=A$. 1. There is a subspace $V$ of co-dimension at most $A^2$ such that $f|_V$ is constant. 2. f can be computed by a parity decision tree of size $2^{A^2}n^{2A}$. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth $A^2 \log s$. 4. For every $0<\epsilon$ there is a parity decision tree of depth $O(A^2 + \log(1/\epsilon))$ and size $2^{O(A^2)} \cdot \min\{1/\epsilon^2,O(\log(1/\epsilon))^{2A}\}$ that \epsilon-approximates f. Furthermore, this tree can be learned, with probability $1-\delta$, using $\poly(n,\exp(A^2),1/\epsilon,\log(1/\delta))$ membership queries. All the results above also hold (with a slight change in parameters) to functions $f:Z_p^n\to \{0,1\}$.