Spectral Norm of Symmetric Functions

Abstract

The spectral norm of a Boolean function $f:\{0,1{\}}^n\to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log (n/r(f))$ where $r(f)=\max \{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x)\cdot parity(x)$ is constant for all $x$ with $\sum x_i\in [r_0,n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions.

References in corpus (1)

• Communication Complexities of XOR functions

Cited by in corpus (1)

• Fourier sparsity, spectral norm, and the Log-rank conjecture