Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications

Abstract

In this paper, convergence results in a multivariate setting have been proved for a family of neural network operators of the maxproduct type. In particular, the coefficients expressed by Kantorovich type means allow to treat the theory in the general frame of the Orlicz spaces, which includes as particular case the L p -spaces. Examples of sigmoidal activation functions are discussed, for the above operators in different cases of Orlicz spaces. Finally, concrete applications to real world cases have been presented in both uni-variate and multivariate settings. In particular, the case of reconstruction and enhancement of biomedical (vascular) image has been discussed in details.2010 Mathematics Subject Classification. 41A25, 41A05, 41A30, 47A58.

Asymptotic Expansion for Neural Network Operators of the Kantorovich Type and High Order of Approximation

“…First of all, we consider the case of the Kantorovich NN type operators activated by the well-known logistic function (see, e.g., [20,40,41] and Fig. 1):…”

New approximation properties of the Bernstein max-min operators and Bernstein max-product operators

“…As an alternative to the well-known linear approximation operators, recently, various types of nonlinear approximation operators have been introduced. First, we mention the so called max-product type operators, a class of subadditive and positively homogeneous operators, used as alternatives for the linear counterparts in many areas such as, approximation operators (see, e. g., [2], [3], [20]), interpolation operators (see, e. g., [5], [9]), sampling operators (see, e. g., [6], [8]), neural network operators (see, e. g., [1], [13], [14]) and others (see, e. g. [18], [17], [19], [21]). A detailed account on the theory of max-product type operators can be found in the monograph [4].…”

On a Durrmeyer-type modification of the Exponential sampling series

“…This represents a wide field of investigations, due to its practical applications in various sectors of applied sciences (see e.g. [1,2,[20][21][22][23][24][25] and references therein).…”