Saturation Classes for Max-Product Neural Network Operators Activated by Sigmoidal Functions

Costarelli, Danilo; Vinti, Gianluca

doi:10.1007/s00025-017-0692-6

Cited by 38 publications

(23 citation statements)

References 38 publications

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“…where the function (x) + := max {x, 0} denotes the positive part of x ∈ R. The corresponding multivariate spline kernels (see e.g., Fig. 2) are then defined by: For others useful examples of kernel functions, see e.g., [12,28,29,30]. Now, we are able to recall the definition of the multivariate sampling Kantorovich operators ( [24]), of the form:…”

Section: Approximation By Sampling Kantorovich Operators and Applicatmentioning

confidence: 99%

Detection of thermal bridges from thermographic images by means of image processing approximation algorithms

Asdrubali

Baldinelli

Bianchi

et al. 2018

Applied Mathematics and Computation

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Section: Approximation By Sampling Kantorovich Operators and Applicatmentioning

confidence: 99%

Detection of thermal bridges from thermographic images by means of image processing approximation algorithms

Asdrubali

Baldinelli

Bianchi

et al. 2018

Applied Mathematics and Computation

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“…In Section 2, the definition of kernel for the sampling Kantorovich operators S w (and also for G w ) has been provided. Several examples of well-known functions χ which satisfy assuptions (χ1), (χ2), and (χ3) are given e.g., in [11,3,20,25,26]. For instance, we can choose as kernels the following one-dimensional band-limited functions: where α > (n − 1)/2, B λ is the Bessel function of order λ and Γ is the Euler function, and finally,…”

Section: The Construction Of the Kernelsmentioning

confidence: 99%

An Inverse Result of Approximation by Sampling Kantorovich Series

Costarelli¹,

Vinti²

2018

Proceedings of the Edinburgh Mathematical Society

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In the present paper, an inverse result of approximation, i.e., a saturation theorem for the sampling Kantorovich operators is derived, in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, here we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by P.L. Butzer. At the end, some other applications of such representation formula are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

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“…For any fixed sigmoidal function σ(x), we can define the following nonnegative density function ( [4,25,26,28]):…”

Section: Sigmoidal Functions and Basic Assumptionsmentioning

confidence: 99%

Approximation Results in Orlicz Spaces for Sequences of Kantorovich Max-Product Neural Network Operators

Costarelli

Sambucini

2018

Results Math

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In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces L ϕ . The results here proved, extend those given by the authors in Result Math., 2016, to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies in the possibility to approximate not necessarily continuous functions (data) belonging to different function spaces by a unique general approach. Further, in order to derive quantitative estimates in this context, we introduce a suitable K-functional in L ϕ and use it to provide an upper bound for the approximation error of the above operators. Finally, examples of sigmoidal activation functions have been considered and studied.1991 Mathematics Subject Classification. Primary: 41A25, 41A05, 41A30, 47A58.

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Saturation Classes for Max-Product Neural Network Operators Activated by Sigmoidal Functions

Cited by 38 publications

References 38 publications

Detection of thermal bridges from thermographic images by means of image processing approximation algorithms

Detection of thermal bridges from thermographic images by means of image processing approximation algorithms

An Inverse Result of Approximation by Sampling Kantorovich Series

Approximation Results in Orlicz Spaces for Sequences of Kantorovich Max-Product Neural Network Operators

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