Harmonic Analysis of Polynomial Threshold Functions

Abstract

The analysis of linear threshold Boolean functions has recently attracted the attention of those interested in circuit complexity as well as of those interested in neural networks. Here a generalization of linear threshold functions is defined, namely, polynomial threshold functions, and its relation to the class of linear threshold functions is investigated. A Boolean function is polynomial threshold if it can be represented as a sign function of a polynomial that consists of a polynomial (in the number of variables) number of terms. The main result of this paper is showing that the class of polynomial threshold functions (which is called $PT_1 $) is strictly contained in the class of Boolean functions that can be computed by a depth 2, unbounded fan-in polynomial size circuit of linear threshold gates (which is called $LT_2 $). Harmonic analysis of Boolean functions is used to derive a necessary and sufficient condition for a function to be an S-threshold function for a given set S of monomials. This condition is used to show that the number of different S-threshold functions, for a given S, is at most $2^{( n + 1 )| S |}$. Based on the necessary and sufficient condition, a lower bound is derived on the number of terms in a threshold function. The lower bound is expressed in terms of the spectral representation of a Boolean function. It is found that Boolean functions having an exponentially small spectrum are not polynomial threshold. A family of functions is exhibited that has an exponentially small spectrum; they are called “semibent” functions. A function is constructed that is both semibent and symmetric to prove that $PT_1 $ is properly contained in $LT_2 $.