On the Structure of Boolean Functions with Small Spectral Norm

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 an affine 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 at most ${2^{A^2} n^{2A}}$. (A parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.)

• 3. f can be computed by a De Morgan formula of size ${O(2^{A^2} n^{2A + 2})}$ and by a De Morgan formula of depth ${O(A^2 + \log(n) \cdot A)}$.

• 4. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth at most ${A^2 \log s}$.

• 5. For every ${\epsilon > 0}$ there is a parity decision tree of depth ${O(A^2 + \log(1/\epsilon))}$ and size ${2^{O(A^2)} \cdot \min \{1/\epsilon^2, \log(1/\epsilon)^{2A}\}}$ that ${\epsilon}$-approximates f. Furthermore, this tree can be learned (in the uniform distribution model), with probability ${1 - \delta}$, using ${{\tt poly}(n, {\rm exp}(A^2), 1/\epsilon, \log(1/\delta))}$ membership queries.

All the results above (except 3) also hold (with a slight change in parameters) for functions ${f : \mathbb{Z}_p^n \to \{0, 1\}}$.