Hyperplane Arrangements Separating Arbitrary Vertex Classes in n-Cubes

Strictly layered feedforward networks with binary neurons are viewed as maps from the vertex set of an n-cube to the vertex set of an l-cube. With only one output neuron, they can in principle realize any Boolean function on n inputs. We address the problem of determining the necessary and sufficient numbers of hidden units for this task by using separability properties of affine oriented hyperplane arrangements.

[1]  Anders Krogh,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[2]  Julian Morris,et al.  A procedure for determining the topology of multilayer feedforward neural networks , 1994, Neural Networks.

[3]  Masahiko Arai,et al.  Bounds on the number of hidden units in binary-valued three-layer neural networks , 1993, Neural Networks.

[4]  A. Barron.  Approximation and Estimation Bounds for Artificial Neural Networks , 1991, COLT '91.

[5]  Kurt Hornik,et al.  Approximation capabilities of multilayer feedforward networks , 1991, Neural Networks.

[6]  H. White,et al.  Universal approximation using feedforward networks with non-sigmoid hidden layer activation functions , 1989, International 1989 Joint Conference on Neural Networks.

[7]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[8]  Ken-ichi Funahashi,et al.  On the approximate realization of continuous mappings by neural networks , 1989, Neural Networks.

[9]  James L. McClelland,et al.  Parallel Distributed Processing: Explorations in the Microstructures of Cognition , 1987, Language.

[10]  James L. McClelland,et al.  Parallel distributed processing: explorations in the microstructure of cognition, vol. 1: foundations , 1986 .

[11]  Geoffrey E. Hinton,et al.  Learning internal representations by error propagation , 1986 .

[12]  T. Zaslavsky.  Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes , 1975 .

[13]  Yih-Fang Huang,et al.  Bounds on the number of hidden neurons in multilayer perceptrons , 1991, IEEE Trans. Neural Networks.

[14]  Robert Hecht-Nielsen,et al.  Theory of the backpropagation neural network , 1989, International 1989 Joint Conference on Neural Networks.

[15]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..