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Spectral Norm of Symmetric Functions

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.