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Problems of Information Transmission

, Volume 45, Issue 1, pp 46–53 | Cite as

Perceptrons of large weight

  • V. V. Podolskii
Automata Theory

Abstract

A threshold gate is a linear combination of input variables with integer coefficients (weights). It outputs 1 if the sum is positive. The maximum absolute value of the coefficients of a threshold gate is called its weight. A degree-d perceptron is a Boolean circuit of depth 2 with a threshold gate at the top and any Boolean elements of fan-in at most d at the bottom level. The weight of a perceptron is the weight of its threshold gate.

For any constant d ≥ 2 independent of the number of input variables n, we construct a degree-d perceptron that requires weights of at least \( n^{\Omega (n^d )} \); i.e., the weight of any degree-d perceptron that computes the same Boolean function must be at least \( n^{\Omega (n^d )} \). This bound is tight: any degree-d perceptron is equivalent to a degree-d perceptron of weight \( n^{O(n^d )} \). For the case of threshold gates (i.e., d = 1), the result was proved by Håstad in [2]; we use Håstad’s technique.

Keywords

Serpentine Boolean Function Information Transmission Large Weight Order Number 
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

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsLomonosov Moscow State UniversityMoscowRussia

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