Proceedings Article10.1109/IJCNN.1998.685957
The general approximation theorem
Alexander N. Gorban,Donald C. Wunsch +1 more
- 04 May 1998
- Vol. 2, pp 1271-1274
TL;DR: A general approximation theorem is proved, which uniformly envelopes both the classical Stone theorem and approximation of functions of several variables by means of superpositions and linear combinations of function of one variable.
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Abstract: A general approximation theorem is proved. It uniformly envelopes both the classical Stone theorem and approximation of functions of several variables by means of superpositions and linear combinations of functions of one variable. This theorem is interpreted as a statement on universal approximating possibilities ("approximating omnipotence") of arbitrary nonlinearity. For the neural networks, our result states that the function of neuron activation must be nonlinear, and nothing else.
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References
Multilayer feedforward networks are universal approximators
TL;DR: It is rigorously established that standard multilayer feedforward networks with as few as one hidden layer using arbitrary squashing functions are capable of approximating any Borel measurable function from one finite dimensional space to another to any desired degree of accuracy, provided sufficiently many hidden units are available.
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Approximation by superpositions of a sigmoidal function
TL;DR: It is demonstrated that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real variables with support in the unit hypercube.