Universal approximation theorem: Difference between revisions

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In mathematics, the '''universal approximation theorem''' claimsstates<ref>Balázs Csanád Csáji. Approximation with Artificial Neural Networks; Faculty of Sciences; Eötvös Loránd University, Hungary</ref> that the standard multilayer feed-forward networksnetwork with a single hidden layer that contains finite number of hidden neurons[[neuron]]s, and with arbitrary activation function are universal approximators in ''C''(Rm). Kurt Hornik (1991) showed that it is not the specific choice of the activation function, but rather the multilayer feedforward architecture itself which gives neural networks the potential of being universal approximators. The output units are always assumed to be linear. For notational convenience we shall explicitly formulate our results only for the case where there is only one output unit. (The general case can easily be deduced from the simple case.) The theorem<ref>G. Cybenko. Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems, 2:303–314, 1989.</ref><ref>

Kurt Hornik: Approximation Capabilities of Multilayer Feedforward Networks.

Neural Networks, vol. 4, 1991.</ref><ref>Haykin, Simon (1998). Neural Networks: A Comprehensive Foundation, 2, Prentice Hall. ISBN 0132733501.</ref> in mathematical terms:

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<i>Let φ(·) be a nonconstant, bounded, and monotome-increasing continuous function. Let ''I''_''m''₀ denote the ''m''₀-dimensional unit hypercube [0,1]^''m''₀. The space of continuous functions on ''I''_''m''₀ is denoted by ''C''(''I''_''m''₀). Then, given any function ''f'' Э ''C''(''I''_''m''₀) and є &gt; 0, there exist an integer ''m''₁ and sets of real constants ''α''_''i'', ''b''_''i'' and ''w''_''ij'', where ''i'' = 1, ..., ''m''₁ and ''j'' = 1, ..., ''m''₀ such that we may define:</i>

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<i>as an approximate realization of the function ''f(·)''; that is,</i></div>

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for all ''x''₁, ''x''₂, ..., ''x''_''m''₀ that lie in the input space.</i>

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