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- W2949532904 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 a 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 $2^{A^2}n^{2A}$. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth $A^2 log s$. 4. For every $0<epsilon$ there is a parity decision tree of depth $O(A^2 + log(1/epsilon))$ and size $2^{O(A^2)} cdot min{1/epsilon^2,O(log(1/epsilon))^{2A}}$ that epsilon-approximates f. Furthermore, this tree can be learned, with probability $1-delta$, using $poly(n,exp(A^2),1/epsilon,log(1/delta))$ membership queries. All the results above also hold (with a slight change in parameters) to functions $f:Z_p^nto {0,1}$." @default.
- W2949532904 created "2019-06-27" @default.
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- W2949532904 date "2013-04-01" @default.
- W2949532904 modified "2023-09-27" @default.
- W2949532904 title "On the Structure of Boolean Functions with Small Spectral Norm" @default.
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