Polynomial Threshold Functions, AC^0 Functions, and Spectral Norms

We study the class of polynomial threshold functions using harmonic analysis and apply the results to derive lower bounds related to ACo functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). Our main result is that the class of polynomial threshold functions can be characterized using their spectral representation. In particular we prove that a Boolean function whose L1 spectral norm is bounded by a polynomial in n is a polynomial threshold function. While a Boolean function which its L;' spectral norm is not bounded by a polynomial in n is not a polynomial threshold function [ 5 ] . When we consider ACo functions, we obtain the following: (i) There exists an ACo function whose L1 spectral norm is exponentially large. (ii) There exists an ACo function whose ~5 ; ' spectral norm is R(nPo'Y1og(n)). Hence, applying our characterization results, we derive 'IBM Almaden Research Center, 650 Harry Road, San 'Computer Science Department, University of Jose, CA 95120-6099. Toronto, Toronto, Canada M5S 1A4. an R(npo'ylog(n)) lower bound on the number of terms needed to compute exactly a Boolean ACO function as a sign function of a polynomial. This result complements the results of [12]. (iii) A lower bound of Q(nJ'o'Y1og(n)) on the size of a depth-2 circuit of MAJORITY gates that computes an ACo function. This is the first known lower bound for the result of [l].