Compactness of Neural Networks

Keiichi Miyajima; Hiroshi Yamazaki

doi:10.2478/forma-2022-0002

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Compactness of Neural Networks

Formalized Mathematics

Volume 30 (2022): Issue 1 (April 2022)

By: Keiichi Miyajima and Hiroshi Yamazaki

Open Access

|Dec 2022

Abstract

In this article, Feed-forward Neural Network is formalized in the Mizar system [1], [2]. First, the multilayer perceptron [6], [7], [8] is formalized using functional sequences. Next, we show that a set of functions generated by these neural networks satisfies equicontinuousness and equiboundedness property [10], [5]. At last, we formalized the compactness of the function set of these neural networks by using the Ascoli-Arzela’s theorem according to [4] and [3].

References

[1] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.
Open DOI Search in Google Scholar Back to article
[2] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.604425130069070
Open DOI Search in Google Scholar Back to article
[3] Serge Lang. Real and Functional Analysis (Texts in Mathematics). Springer-Verlag, 1993.10.1007/978-1-4612-0897-6
Search in Google Scholar Back to article
[4] Kazuo Matsuzaka. Sets and Topology (Introduction to Mathematics). IwanamiShoten, 2000.
Search in Google Scholar Back to article
[5] Michael Read and Barry Simon. Functional Analysis (Methods of Modern Mathematical Physics). Academic Press, 1980.
Search in Google Scholar Back to article
[6] Frank Rosenblatt. The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain. Psychological Review, 1958.10.1037/h004251913602029
Search in Google Scholar Back to article
[7] David Everett Rumelhart, Geoffrey Everes Hinton, and Ronald J. Williams. Learning representations by backpropagating errors. Nature, 1986.10.1038/323533a0
Search in Google Scholar Back to article
[8] Jürgen Schmidhuber. Deep Learning in Neural Networks: An Overview. Neural Networks, 2015.10.1016/j.neunet.2014.09.00325462637
Search in Google Scholar Back to article
[9] Hiroshi Yamazaki, Keiichi Miyajima, and Yasunari Shidama. Ascoli-Arzelà theorem. Formalized Mathematics, 29(2):87–94, 2021. doi:10.2478/forma-2021-0009.
Open DOI Search in Google Scholar Back to article
[10] Kôsaku Yosida. Functional Analysis. Springer, 1980.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/forma-2022-0002 | Journal eISSN: 1898-9934 | Journal ISSN: 1426-2630

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Language: English

Page range: 13 - 21

Accepted on: Apr 30, 2022

Published on: Dec 21, 2022

Published by: University of Białystok

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

neural network,

compactness,

Ascoli-Arzela’s theorem,

equicontinuousness of continuous functions,

equiboundedness of continuous functions

Related subjects:

Computer sciences,

Computer sciences, other,

Mathematics,

General mathematics

© 2022 Keiichi Miyajima, Hiroshi Yamazaki, published by University of Białystok
This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 License.

Volume 30 (2022): Issue 1 (April 2022)