Multivariate neural network operators with sigmoidal activation functions

• Published in: Neural networks Vol. 48; pp. 72 - 77
• Main Authors: Costarelli, Danilo, Spigler, Renato
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.12.2013; Elsevier
• Subjects: Activation; Activation analysis; Algorithms; Applied sciences; Approximation; Artificial intelligence; Computer science; control theory; systems; Computing Methodologies; Connectionism. Neural networks; Convergence; Exact sciences and technology; Linear Models; Lipschitz classes; Logistics; Mathematical analysis; Multivariate Analysis; Multivariate neural networks operators; Neural networks; Neural Networks (Computer); Operators; Order of approximation; Programming, Linear; Sigmoidal functions; Uniform approximation; Lipschitz classes; Multivariate neural networks operators; Sigmoidal functions; Order of approximation; Uniform approximation; Uniform convergence; Multivariate interpolation; Activation function; Neural network; Sigmoidal transformation; Logistics
• ISSN: 0893-6080; 1879-2782
• DOI: 10.1016/j.neunet.2013.07.009

In this paper, we study pointwise and uniform convergence, as well as order of approximation, of a family of linear positive multivariate neural network (NN) operators with sigmoidal activation functions. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of NN operators, activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. Multivariate NNs approximation finds applications, typically, in neurocomputing processes. Our approach to NN operators allows us to extend previous convergence results and, in some cases, to improve the order of approximation. The case of multivariate quasi-interpolation operators constructed with sigmoidal functions is also considered.