1,721,004 research outputs found
Approximation error for neural network operators by an averaged modulus of smoothness
No full textIn the present paper we establish estimates for the error of approximation (in the L p-norm) achieved by neural network (NN) operators. The above estimates have been given by means of an averaged modulus of smoothness introduced by Sendov and Popov, also known with the name of & tau;-modulus, in case of bounded and measurable functions on the interval [-1, 1]. As a consequence of the above estimates, we can deduce an L p convergence theorem for the above family of NN operators in case of functions which are bounded, measurable, and Riemann integrable on the above interval. In order to reach the above aims, we preliminarily establish a number of results; among them we can mention an estimate for the p-norm of the operators, and an asymptotic type theorem for the NN operators in case of functions belonging to Sobolev spaces.& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Approximation of functions of several variables by deep operators activated by sigmoidal functions and rectified power units
No full textIn the present paper the constructive theory of multivariate deep operators activated by sigmoidal functions is introduced and studied. The main result here establishes a pointwise and uniform convergence theorem in the space of continuous functions on the compact [-1,1]d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}[-1,1]<^>d\end{document}, d >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. Moreover, we also prove quantitative estimates for the order of approximation and the corresponding qualitative rate of convergence. In order to achieve the above theorems, some auxiliary results have been preliminary proved. The above introduced operators belong to the field of positive linear operators, and they also represent a sort of deep (multi-layer) counterpart of the well-known (shallow) operators activated by sigmoidal functions studied in recent years. The above family of deep operators can be considered activated also by the ReLU and powers of ReLU, also known as rectified linear unit and rectified powers units, respectively
A collocation method for solving nonlinear Volterra integro-differential equations of the neutral type by sigmoidal functions
No full textA numerical collocation method is developed for solving nonlinear Volterra
integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are
obtained using unit step functions, and important applications can be obtained
also using other sigmoidal functions, such as logistic and Gompertz functions.
The method allows to obtain a {\em simultaneous} approximation of the solution
to a given VIDE and its first derivative, by means of an explicit formula. 'A priori' as well as 'posteriori' estimates are derived for the
numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial
collocation method as for accuracy and CPU time
Multivariate Neural Network Operators: Simultaneous Approximation and Voronovskaja-Type Theorem
No full textIn this paper, the simultaneous approximation and a Voronoskaja-type theorem for the multivariate neural network operators of the Kantorovich type have been proved. In order to establish such results, a suitable multivariate Strang-Fix type condition has been assumed. A crucial step in the established proofs is given by the application of certain auxiliary results (here established) involving the partial derivatives of the considered multivariate density functions. Other than convergence theorems, we also establish quantitative estimates for the order of simultaneous approximation thanks to the use of the modulus of continuity of the target function. Here, sigmoidal, rectified linear unit (ReLu), and rectified power units (RePUs) functions have been considered as activation functions
Best Approximation and Inverse Results for Neural Network Operators
No full textIn the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail
Approximation by series of sigmoidal functions with applications to neural networks
No full textIn this paper, we develop a constructive theory for approximating absolutely
continuous functions by series of certain sigmoidal functions. Estimates for the approximation error are also derived. The relation with neural networks (NNs) approximation is discussed. The connection between sigmoidal functions and the scaling functions of -regular multiresolution approximations are investigated. In this setting, we show that the approximation error for -functions decreases as , as . Examples with sigmoidal functions of several kinds, such as logistic, hyperbolic tangent, and Gompertz functions, are given
Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
No full textIn the present work, we establish a quantitative estimate for the perturbed sampling Kantorovich operators in Orlicz spaces, in terms of the modulus of smoothness, defined by means of its modular functional. From the obtained result, we also deduce the qualitative order of approximation, by considering functions in suitable Lipschitz classes. This allows us to apply the above results in certain Orlicz spaces of particular interest, such as the interpolation spaces, the exponential spaces and the L-p -spaces, 1 <= p < + infinity. In particular, in the latter case, we also provide an estimate established using a direct proof based on certain properties of the L-p -modulus of smoothness, which are not valid in the general case of Orlicz spaces. The possibility of using a direct approach allows us to improve the estimate that can be deduced as a consequence of the one achieved in Orlicz spaces. In the final part of the article, we furnish some estimates and the corresponding qualitative order of approximation in the space of uniformly continuous and bounded functions
How sharp is the Jensen inequality?
No full textWe study how good is Jensen's inequality, that is the discrepancy between , and , being convex and a nonnegative function. Such an estimate can be useful to provide error bounds for certain approximations in , or in Orlicz spaces, where convex modular functionals are often involved. Estimates for the case of functions, as well as for merely Lipschitz continuous convex functions , are established. Some examples are given to illustrate how sharp our results are, and a comparison is made with some other estimates existing in the literature. Finally, some
applications involving the Gamma function are obtained.We study how good the Jensen inequality is, that is, the discrepancy between (Formula presented.) , φ being convex and f(x) a nonnegative L1 function. Such an estimate can be useful to provide error bounds for certain approximations in Lp, or in Orlicz spaces, where convex modular functionals are often involved. Estimates for the case of C2 functions, as well as for merely Lipschitz continuous convex functions φ, are established. Some examples are given to illustrate how sharp our results are, and a comparison is made with some other estimates existing in the literature. Finally, some applications involving the Gamma function are obtained
Convergence of a family of neural network operators of the Kantorovich type
No full textA family of neural network operators of the Kantorovich type is introduced and their convergence studied. Such operators are multivariate, and based on certain special density functions, constructed through sigmoidal functions. Pointwise as well as uniform approximation theorems are established when such operators are applied to continuous functions. Moreover, also approximations are considered, with 1 \miu p < +\infty, since the setting is the most natural for the neural network operators of the Kantorovich type. Constructive multivariate approximation algorithms, based on neural networks, are important since typical applications to neurocomputing processes do exist for high-dimensional data, then the relation with usual neural networks
approximations is discussed. Several examples of sigmoidal functions, for which the present theory can be applied are presented
Approximation results in Sobolev and fractional Sobolev spaces by sampling Kantorovich operators
No full textThe present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in "Sobolev-like settings". More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type conditions have been required on the kernel functions defining the above sampling type series. As a consequence, certain open problems related to the convergence in variation for the SK operators have been solved. Then, we considered the above operators in a fractional-type setting. It is well-known that, in the literature, several notions of fractional Sobolev spaces are available, such as, the Gagliardo Sobolev spaces (GSs) defined by means of the Gagliardo semi-norm, or the weak Riemann-Liouville Sobolev spaces (wRLSs) defined by the weak (left and right) Riemann-Liouville fractional derivatives and so on. Here, in order to face the above convergence problem, we introduced a new definition of fractional Sobolev spaces, that we called the tight fractional Sobolev spaces (tfSs) and generated as the intersection of the GSs and the symmetric Sobolev spaces (i.e., that given by the intersection of the left and the right wRLSs). In the latter setting, we obtain one of the main results of the paper, that is a convergence theorem for the SK operators with respect to a suitable norm on tfSs
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