On the Structure of Boolean Functions with Small Spectral Norm

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is ‖ f^ ‖ ₁= ∑ _α| f^ (α) |). Specifically, we prove the following results for functions f: { 0 , 1 } ⁿ→ { 0 , 1 } with ‖ f^ ‖ ₁= A.1.There is an affine subspace V of co-dimension at most A² such that f| _V is constant.2.f can be computed by a parity decision tree of size at most 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(2A2n2A+2) and by a De Morgan formula of depth 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 A²log s.5.For every ϵ> 0 there is a parity decision tree of depth O(A²+ log (1 / ϵ)) and size 2O(A2)·min{1/ϵ2,log(1/ϵ)2A} that ϵ-approximates f. Furthermore, this tree can be learned (in the uniform distribution model), with probability 1 - δ, using 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:Zpn→{0,1}.