Introduction to the theory of neural computation
Introduction to the theory of neural computation
Circuit complexity and neural networks
Circuit complexity and neural networks
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
Geometric sets of low information content
Theoretical Computer Science
Anti-Hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs
Journal of Combinatorial Theory Series A
A complex-number fourier technique for lower bounds on the mod-m degree
Computational Complexity
On Probabilistic ACC Circuits with an Exact-Threshold Output Gate
ISAAC '92 Proceedings of the Third International Symposium on Algorithms and Computation
Problems of Information Transmission
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
A uniform lower bound on weights of perceptrons
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Dimension, halfspaces, and the density of hard sets
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Computing symmetric boolean functions by circuits with few exact threshold gates
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
A better upper bound on weights of exact threshold functions
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
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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We consider Boolean exact threshold functions defined by linear equations, and in general degree d polynomials. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close and in the linear case in particular they are almost matching. The quantity is the same as the maximum magnitude of integer coefficients of linear equations required to express every possible intersection of a hyperplane in Rn and the Boolean cube {0, 1}n, or in the general case intersections of hypersurfaces of degree d in Rn and the Boolean cube {0, 1}n. In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension k of the affine subspace spanned by the solutions, and give upper and lower bounds in this case as well. Our bounds here in terms of k leave a substantial gap, a challenge for future work.