On the Structure of Boolean Functions with Small Spectral Norm

AbstractIn this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is $${\|\hat{f}\|_1 = \sum_{\alpha}|\hat{f}(\alpha)|}$$‖f^‖1=∑α|f^(α)|). Specifically, we prove the following results for functions $${f : \{0, 1\}^n \to \{0, 1\}}$$f:{0,1}n→{0,1} with $${\|\hat{f}\|_1 = A}$$‖f^‖1=A. 1.There is an affine subspace V of co-dimension at most A2 such that $${f|_V}$$f|V is constant.2.f can be computed by a parity decision tree of size at most $${2^{A^2} n^{2A}}$$2A2n2A. (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 $${O(2^{A^2} n^{2A + 2})}$$O(2A2n2A+2) and by a De Morgan formula of depth $${O(A^2 + \log(n) \cdot A)}$$O(A2+log(n)·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 $${A^2 \log s}$$A2logs.5.For every $${\epsilon > 0}$$ϵ>0 there is a parity decision tree of depth $${O(A^2 + \log(1/\epsilon))}$$O(A2+log(1/ϵ)) and size $${2^{O(A^2)} \cdot \min \{1/\epsilon^2, \log(1/\epsilon)^{2A}\}}$$2O(A2)·min{1/ϵ2,log(1/ϵ)2A} that $${\epsilon}$$ϵ-approximates f. Furthermore, this tree can be learned (in the uniform distribution model), with probability $${1 - \delta}$$1-δ, using $${{\tt poly}(n, {\rm exp}(A^2), 1/\epsilon, \log(1/\delta))}$$poly(n,exp(A2),1/ϵ,log(1/δ)) membership queries. All the results above (except 3) also hold (with a slight change in parameters) for functions $${f : \mathbb{Z}_p^n \to \{0, 1\}}$$f:Zpn→{0,1}.