Quantitative estimates involving K-functionals for neural network-type operators

ABSTRACT

In the present paper, quantitative estimates for the neural network (NN) operators of the Kantorovich type have been proved. Firstly, the modulus of continuity of the function being approximated has been used in order to estimate the approximation error in the uniform norm. Finally, a Peetre K-functional has been employed to obtain a quantitative upper bound in $L^p$-norm. At the end, several examples of sigmoidal activation functions for the above NN-type operators have been provided.

KEYWORDS:

• Sigmoidal function
• neural networks operator
• K-functional
• modulus of continuity
• rate of approximation
• quantitative estimate

AMS SUBJECT CLASSIFICATIONS:

• 41A25
• 41A05
• 41A30
• 47A58

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are partially supported by the ‘Department of Mathematics and Computer Science’ of the University of Perugia (Italy). Moreover, the first author of the paper holds a research grant (Post-Doc) funded by the INdAM, and finally, he has been partially supported within the 2017 GNAMPA-INdAM Project ‘Approssimazione con operatori discreti e problemi di minimo per funzionali del calcolo delle variazioni con applicazioni all’imaging’.