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

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.

Some applications of modular convergence in vector lattice setting

“…In this paper we continue the study started in [3,10,11,[16][17][18][19] were we have extended to the vector lattice setting the problem of approximating a function f by means of Urysohn-type integral operators in the setting of modular convergence. These operators are particularly useful in order to approximate a continuous or analog signal by means of discrete samples, and therefore they are widely applied for instance in reconstructing images, see for example [1,2,5,6,[29][30][31][32]. Professor Paul Leo Butzer is certainly one of the Masters and pioneers in Approximation Theory and Signal Analysis and his works have formed many generations of researchers.…”

Abstract Integration with Respect to Measures and Applications to Modular Convergence in Vector Lattice Setting

“…In the literature there are many studies concerning the problem of approximating a real-valued function f by Urysohn-type integral operators or discrete sampling operators. These topics together with some other kind of operators, have several applications in several branches, for instance neural networks and reconstruction of signals and images (see, e.g., [1,2,[5][6][7][8][9]22,[25][26][27][28][29]).…”