On the Size of Weights for Threshold Gates
SIAM Journal on Discrete Mathematics
Perceptrons, PP, and the polynomial hierarchy
Computational Complexity - Special issue on circuit complexity
On Computation and Communication with Small Bias
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Problems of Information Transmission
A uniform lower bound on weights of perceptrons
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
SIAM Journal on Computing
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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 nΩ(nd), that is, the weight of any perceptron of order d computing the same Boolean function is at least nΩ(nd). This bound is tight: every perceptron of order d is equivalent to a perceptron of order d and weight nΩ(nd). In the case of threshold gates (i.e. d = 1) the result was established by Håstad in [1]; we use Håstad's techniques.