A Uniform Lower Bound on Weights of Perceptrons

  • Vladimir V. Podolskii
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)


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 Open image in new window we construct a function computable by a perceptron of order d but not computable by any perceptron of order D with total weight \(2^{o(n^d/D^{4d})}\). In particular, if D is a constant, our function is not computable by any perceptron of order D with total weight \(2^{o(n^d)}\). Previously functions with this properties were known only for d = 1 (and arbitrary D) [2] and for D = d [12].


Total Weight Boolean Function Maximal Element Ordinal Number Minimal Pair 
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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Vladimir V. Podolskii
    • 1
  1. 1.Moscow State University 

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