Inverse result of approximation for the max-product neural network operators of the kantorovich type and their saturation order

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.