A uniform lower bound on weights of perceptrons

Abstract

A threshold gate is a linear function of input variables with integer coefficients (weights). It outputs 1 if the value of the function is positive. The sum of absolute values of coefficients is called the total weight of the threshold gate. A perceptron of order d is a circuit of depth 2 having a threshold gate on the top level and conjunctions of fan-in at most d on the remaining level. For every n and we construct a function computable by a perceptron of order d but not computable by any perceptron of order D with total weight . In particular, if D is a constant, our function is not computable by any perceptron of order D with total weight . Previously functions with this properties were known only for d∈=∈1 (and arbitrary D) [2] and for D∈=∈d [12]. © 2008 Springer-Verlag Berlin Heidelberg.