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

Abstract

In this paper, we study the rate of pointwise approximation for the neural network operators of the Kantorovich type. This result is obtained proving a certain asymptotic expansion for the above operators and then by establishing a Voronovskaja type formula. A central role in the above resuts is played by the truncated algebraic moments of the density functions generated by suitable sigmoidal functions. Furthermore, to improve the rate of convergence, we consider finite linear combinations of the above neural network type operators, and also in the latter case, we obtain a Voronovskaja type theorem. Finally, concrete examples of sigmoidal activation functions have been deeply discussed, together with the case of rectified linear unit (ReLu) activation function, very used in connection with deep neural networks.

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

“…are also preserved. Problems of interpolation, or more in general, of approximation, are related to the topic of training a neural network by sample values belonging to a certain training set: this explains the interest in studying approximation results by means of NN operators in various contexts [15,[20][21][22][23][24].…”