Inverse Result of Approximation for the Max-Product Neural Network Operators of the Kantorovich Type and Their Saturation Order

Marco Cantarini, Lucian Coroianu, Danilo Costarelli, Sorin G. Gal and Gianluca Vinti
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Marco Cantarini: Department of Industrial Engineering and Mathematical Sciences, Marche Polytechnic University, 60121 Ancona, Italy
Lucian Coroianu: Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania
Danilo Costarelli: Department of Mathematics and Computer Science, University of Perugia, 06123 Perugia, Italy
Sorin G. Gal: Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania
Gianluca Vinti: Department of Mathematics and Computer Science, University of Perugia, 06123 Perugia, Italy

Mathematics, 2021, vol. 10, issue 1, 1-11

Abstract: In this paper, we consider the max-product neural network operators of the Kantorovich type based on certain linear combinations of sigmoidal and ReLU activation functions. In general, it is well-known that max-product type operators have applications in problems related to probability and fuzzy theory, involving both real and interval/set valued functions. In particular, here we face inverse approximation problems for the above family of sub-linear operators. We first establish their saturation order for a certain class of functions; i.e., we show that if a continuous and non-decreasing function f can be approximated by a rate of convergence higher than 1 / n , as n goes to + ∞ , then f must be a constant. Furthermore, we prove a local inverse theorem of approximation; i.e., assuming that f can be approximated with a rate of convergence of 1 / n , then f turns out to be a Lipschitz continuous function.

Keywords: sigmoidal functions; ReLU function; neural network operators; saturation result; local inverse theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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