## Abstract

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 A^{2} 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(A^{2}+ 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.5.For every ϵ> 0 there is a parity decision tree of depth O(A^{2}+ 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 (A^{2}) , 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}.

Original language | English |
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Pages (from-to) | 229-273 |

Number of pages | 45 |

Journal | Computational Complexity |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2017 |

## Keywords

- Analysis of Boolean functions
- parity decision trees
- spectral norm