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A Uniform Lower Bound on Weights of Perceptrons

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

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 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].

Keywords

Total Weight Boolean Function Maximal Element Ordinal Number Minimal Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

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

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