By George A. Anastassiou (auth.)
ISBN-10: 3642214304
ISBN-13: 9783642214301
This short monograph is the 1st one to deal completely with the quantitative approximation via man made neural networks to the identity-unit operator. the following we examine with charges the approximation homes of the "right" sigmoidal and hyperbolic tangent man made neural community confident linear operators. specifically we examine the measure of approximation of those operators to the unit operator within the univariate and multivariate circumstances over bounded or unbounded domain names. this is often given through inequalities and with using modulus of continuity of the concerned functionality or its larger order by-product. We study the true and complicated cases.
For the ease of the reader, the chapters of this publication are written in a self-contained style.
This treatise depends on author's final years of similar learn work.
Advanced classes and seminars might be taught out of this short ebook. All useful history and motivations are given in keeping with bankruptcy. A comparable record of references is given additionally according to bankruptcy. The uncovered effects are anticipated to discover purposes in lots of parts of desktop technological know-how and utilized arithmetic, reminiscent of neural networks, clever structures, complexity conception, studying conception, imaginative and prescient and approximation idea, and so on. As such this monograph is acceptable for researchers, graduate scholars, and seminars of the above topics, additionally for all technology libraries.
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Extra info for Intelligent Systems: Approximation by Artificial Neural Networks
Sample text
1 (N − 1)! 1 nα 1 . nαN N ! ≤ x, then cos nk ≥ cos x. Hence cos k n cos cos x cos cos x k n cos k −w n k −w n N −1 F (N ) (w) − F (N ) (cos x) dw ≤ N −1 1 k ω1 F (N ) , cos − cos x (N − 1)! n 1 k ω1 F (N ) , −x (N − 1)! 3 Real Neural Network Quantitative Approximations 1 k ω1 F (N ) , −x N! n k n We proved for k −x n N ≤ 27 1 1 ω1 F (N ) , α N! n 1 nαN . ≤ x that |λ| ≤ ω1 F (N) , 1 nα 1 . nαN 1 nα 1 , nαN N ! So in both cases we got that |λ| ≤ ω1 F (N) , when nk − x ≤ n1α . Also in general ( nk ≥ x case) |λ| ≤ 1 (N − 1)!
Approx. : Approximation by superposition of a sigmoidal function. Adv. Applied Math. : Degree of approximation by neural networks with a single hidden layer. Adv. Applied Math. : Constructive function approximation by three-layer artificial neural networks. : Approximation Theory of Real Functions. : The essential order of approximation for neural networks. Science in China (Ser. F) 47, 97–112 (2004) Chapter 2 Univariate Hyperbolic Tangent Neural Network Quantitative Approximation Here we give the univariate quantitative approximation of real and complex valued continuous functions on a compact interval or all the real line by quasi-interpolation hyperbolic tangent neural network operators.
N j=1 ω1 f (N ) , 1 nα 1 nαN N ! + 2e4 f (N ) ∞ N! 29) . 46 2 Univariate Hyperbolic Tangent Neural Network Proof. Next we apply Taylor’s formula with integral remainder. We have (here nk , x ∈ [a, b]) f k n N = j=0 f (j) (x) j! k n j k −x n + N −1 −t dt. (N − 1)! k n f (N ) (t) − f (N ) (x) x Then N k n f Ψ (nx − k) = j=0 k n Ψ (nx − k) f (j) (x) Ψ (nx − k) j! k −x n j + N −1 −t dt. (N − 1)! k n f (N) (t) − f (N ) (x) x Hence nb f k= na N j=1 nb k n Ψ (nx − k) − f (x) Ψ (nx − k) = k= na f (j) (x) j!
Intelligent Systems: Approximation by Artificial Neural Networks by George A. Anastassiou (auth.)
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