We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows that low-degree polynomial threshold functions cannot approximate any function with many influential variables. We provide a couple of examples where this technique yields tight approximation bounds. Our second result relates to constructing pseudorandom generators fooling low-degree polynomial threshold functions. This problem has received attention recently, where proved that @math -wise independence suffices to fool linear threshold functions. We prove that any low-degree polynomial threshold function, which can be represented as a function of a small number of linear threshold functions, can also be fooled by @math -wise independence. We view this as an important step towards fooling general polynomial threshold functions, and we discuss a plausible approach achieving this goal based on our techniques. Our results combine tools from real approximation theory, hyper-contractive inequalities and probabilistic methods. In particular, we develop several new tools in approximation theory which may be of independent interest.
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 @math ) 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 @math ).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 cond... We consider the problem of approximating a Boolean functionf∶ 0,1 n → 0,1 by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP A is properly contained in PSPACE A . We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.