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Perceptrons of Large Weight

  • Vladimir V. Podolskii
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4649)

Abstract

A threshold gate is a sum of input variables with integer coefficients (weights). It outputs 1 if the sum is positive. The maximal absolute value of coefficients of a threshold gate is called its weight. A perceptron of order d is a circuit of depth 2 having a threshold gate on the top level and any Boolean gates of fan-in at most d on the remaining level.

For every constant d ≥ 2 independent of the number of inputs n we exhibit a perceptron of order d that requires weights at least \(n^{\Omega(n^d)}\), that is, the weight of any perceptron of order d computing the same Boolean function is at least \(n^{\Omega(n^d)}\). This bound is tight: every perceptron of order d is equivalent to a perceptron of order d and weight \(n^{O(n^d)}\). In the case of threshold gates (i.e. d = 1) the result was established by Håstad in [1]; we use Håstad’s techniques.

Keywords

Boolean Function Ordinal Number Bottom Level Polynomial Hierarchy Threshold Gate 
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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References

  1. 1.
    Håstad, J.: On the size of weights for threshold gates. SIAM Journal on Discrete Mathematics 7(3), 484–492 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
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    Muroga, S.: Threshold logic and its applications. Wiley-Interscience, Chichester (1971)zbMATHGoogle Scholar
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    Minsky, M.L., Papert, S.A.: Perceptrons. MIT Press, Cambridge, MA (1968)Google Scholar
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    Beigel, R.: Perceptrons, PP and the polynomial hierarchy. Computational Complexity 4, 339–349 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
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    Buhrman, H., Vereshchagin, N., de Wolf, R.: On computation and communication with small bias. In: Accepted for IEEE Conf. on Computational Compl. (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

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

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