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 ||f̂||1 = Σ ^α |f̂(α)|). Specifically, we prove the following results for functions f: {0, 1}ⁿ → {0,1} with ||f̂||1 = A. 1. There is a 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 2 ^A2n^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^A2n^2A+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 A² log s. 5. For every ε > 0 there is a parity decision tree of depth O(A² + log(1/ε)) and size 2 ^O(A2) · min{1/ε², O(log(1/ε)) ^2A} that e-approximates f. Furthermore, this tree can be learned, 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: ℤ_p ⁿ → {0,1}.