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
Perceptrons: An Introduction to Computational Geometry
Perceptrons: An Introduction to Computational Geometry
On Computation and Communication with Small Bias
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
A uniform lower bound on weights of perceptrons
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
Weights of exact threshold functions
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Lower bound on weights of large degree threshold functions
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
Discrete Applied Mathematics
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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.