Spectral norm of symmetric functions

Abstract

The spectral norm of a Boolean function f : {0,1} n →{-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)·PARITY(x) is constant for all x with Σx i ∈[r 0, n - r 1]. We mention some applications to the decision tree and communication complexity of symmetric functions. © 2012 Springer-Verlag.