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Revision #1 to TR13-049 | 22nd May 2013 20:30

#### On the Structure of Boolean Functions with Small Spectral Norm

Revision #1
Authors: Amir Shpilka, Avishay Tal, Ben lee Volk
Accepted on: 22nd May 2013 20:30
Keywords:

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 $f:Z_p^n\to \{0,1\}$.

### Paper:

TR13-049 | 1st April 2013 15:11

#### On the Structure of Boolean Functions with Small Spectral Norm

TR13-049
Authors: Amir Shpilka, Ben Lee Volk, Avishay Tal
Publication: 1st April 2013 15:15
Keywords:

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 $n^{A^2}$. (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 $A^3\log(1/\epsilon)$ that $\epsilon$-approximates $f$. Furthermore, this tree can be learned, with probability $1-\delta$, using poly$(n,A,1/\epsilon,\log(1/\delta))$ membership queries.

All the results above also hold (with a slight change in parameters) to functions $f:Z_p^n\to \{0,1\}$.

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