Approximation of functions on a compact set by finite sums of a sigmoid function without scaling

Ito, Yoshifusa

doi:10.1016/0893-6080(91)90060-i

Cited by 106 publications

(33 citation statements)

References 5 publications

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“…The proof is based on elementary tools like the TaylorYoung formula and the polynomial approximation. A similary approach was made by Ito in [3] but he gave neither the result for the derivatives nor rates of approximation. 9 forA and g in C(K,91),…”

Section: Introductionmentioning

confidence: 84%

Approximation of functions by perceptrons: a new approach

Attali

Pagès

1995

Neural Process Lett

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Abstract. We provide a constructive proof of the theorem of function approximation by perceptrons (of Leshno et al. [i], Homik [2] ) when the activation function W is C" with all its derivatives non 0 at 0. We deal with uniform approximation on compact sets of continuous functions on 91 a, d> 1. This approach is elementary and provides some approximation results for the derivatives along with some bounds for the hidden layer.

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Section: Introductionmentioning

confidence: 84%

Approximation of functions by perceptrons: a new approach

Attali

Pagès

1995

Neural Process Lett

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“…As an important and active topic in this line of research, the research on the approximation capabilities of single hidden layer feedforward neural networks (SLFNs) has attracted the attention from an increasing number of researchers. Some previous research on this topic can be found in Cybenko [5], Funahashi [7], Hornik et al [9,10,12], Ito [13], Chui and Li [4], Leshno et al [14], Chen et al [1][2][3], Liao et al [15], and so on. These studies show that if the network's activation functions obey an explicit set of assumptions, then the network can indeed be shown to be a universal approximator.…”

Section: Introductionmentioning

confidence: 99%

On a problem of Hornik

Xie

Cao

2015

Acta. Math. Sin.-English Ser.

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In 1991, Hornik proved that the collection of single hidden layer feedforward neural networks (SLFNs) with continuous, bounded, and non-constant activation function σ is dense in C(K) where K is a compact set in R s (see Neural Networks, 4(2), 251-257 (1991)). Meanwhile, he pointed out "Whether or not the continuity assumption can entirely be dropped is still an open quite challenging problem". This paper replies in the affirmative to the problem and proves that for bounded and continuous almost everywhere (a.e.) activation function σ on R, the collection of SLFNs is dense in C(K) if and only if σ is un-constant a.e..

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“…In [60], it was shown that, if h : R → R is a continuous function, bounded by a polynomial of degree Other extensions were described in [52], [61], [62], [63], [64], [65], [66], [67], [68] and the references therein. In the papers [69] and [70], the investigations of the C-and L p -density properties were given an exhaustive answer.…”

Section: Being Sigmoidal Is Not Substantialmentioning

confidence: 99%

“…In the papers [51] and [61], various density properties were proved for monotone sigmoidal functions, using only weights with a norm equal to 1. The case of continuous sigmoidal computational units and of weights and thresholds taking only integer values was addressed in [73].…”

Section: Restricting the Parameter Setmentioning

confidence: 99%