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 \(2^{o(n^d/D^{4d})}\). 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].
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Podolskii, V.V. (2008). A Uniform Lower Bound on Weights of Perceptrons. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_27
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DOI: https://doi.org/10.1007/978-3-540-79709-8_27
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