Approximation capability in C(R¯n) by multilayer feedforward networks and related problems

In this paper, we investigate the capability of approximating functions in C(R (n)) by three-layered neural networks with sigmoidal function in the hidden layer. It is found that the boundedness condition on the sigmoidal function plays an essential role in the approximation, as contrast to continuity or monotonity condition. We point out that in order to prove the neural network in the n-dimensional case, all one needs to do is to prove the case for one dimension. The approximation in L(p)-norm (1<p<infinity) and some related problems are also discussed.

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

[2]  Yoshifusa Ito,et al.  Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory , 1991, Neural Networks.

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

[4]  Hong Chen,et al.  Approximations of continuous functionals by neural networks with application to dynamic systems , 1993, IEEE Trans. Neural Networks.

[5]  Halbert White,et al.  Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

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

[7]  S. M. Carroll,et al.  Construction of neural nets using the radon transform , 1989, International 1989 Joint Conference on Neural Networks.

[8]  Yoshifusa Ito,et al.  Approximation of continuous functions on Rd by linear combinations of shifted rotations of a sigmoid function with and without scaling , 1992, Neural Networks.

[9]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[10]  Edward K. Blum,et al.  Approximation theory and feedforward networks , 1991, Neural Networks.

[11]  C. Micchelli,et al.  Approximation by superposition of sigmoidal and radial basis functions , 1992 .

[12]  V. Tikhomirov.  On the Representation of Continuous Functions of Several Variables as Superpositions of Continuous Functions of one Variable and Addition , 1991 .