Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces

Michael Koller; Johannes Grobmann; Ullrich Monich; Holger Boche

doi:10.1109/ICASSP.2018.8462158

Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces

Michael Koller
, Johannes Grobmann
, Ullrich Monich
, Holger Boche

Technical University of Munich
Professur für Methoden der Signalverarbeitung
Lehrstuhl für Theoretische Informationstechnik

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

Abstract

Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs). It was shown that DCNN s are stable with respect to deformations of bandlimited input functions. In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces. Further, we show a weak form of deformation stability for the whole input space L²(Rd). The basic components of DCNNs are semi-discrete frames. For practical applications, a concrete choice is necessary. Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.

Original language	English
Title of host publication	2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	6872-6876
Number of pages	5
ISBN (Print)	9781538646588
DOIs	https://doi.org/10.1109/ICASSP.2018.8462158
State	Published - 10 Sep 2018
Externally published	Yes
Event	2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Calgary, Canada Duration: 15 Apr 2018 → 20 Apr 2018

Publication series

Name	ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings
Volume	2018-April
ISSN (Print)	1520-6149

Conference

Conference	2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018
Country/Territory	Canada
City	Calgary
Period	15/04/18 → 20/04/18

Keywords

Admissible semi -discrete frame
Bounded uniform partition of unity
Deep convolutional neural networks
Deformation stability
Sobolev space

Access to Document

10.1109/ICASSP.2018.8462158

Cite this

Koller, M., Grobmann, J., Monich, U., & Boche, H. (2018). Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces. In 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings (pp. 6872-6876). Article 8462158 (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings; Vol. 2018-April). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ICASSP.2018.8462158

Koller, Michael ; Grobmann, Johannes ; Monich, Ullrich et al. / Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces. 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings. Institute of Electrical and Electronics Engineers Inc., 2018. pp. 6872-6876 (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings).

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title = "Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces",

abstract = "Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs). It was shown that DCNN s are stable with respect to deformations of bandlimited input functions. In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces. Further, we show a weak form of deformation stability for the whole input space L2(Rd). The basic components of DCNNs are semi-discrete frames. For practical applications, a concrete choice is necessary. Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.",

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author = "Michael Koller and Johannes Grobmann and Ullrich Monich and Holger Boche",

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Koller, M, Grobmann, J, Monich, U & Boche, H 2018, Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces. in 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings., 8462158, ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, vol. 2018-April, Institute of Electrical and Electronics Engineers Inc., pp. 6872-6876, 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018, Calgary, Canada, 15/04/18. https://doi.org/10.1109/ICASSP.2018.8462158

Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces. / Koller, Michael; Grobmann, Johannes; Monich, Ullrich et al.
2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings. Institute of Electrical and Electronics Engineers Inc., 2018. p. 6872-6876 8462158 (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings; Vol. 2018-April).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Koller, Michael

AU - Grobmann, Johannes

AU - Monich, Ullrich

AU - Boche, Holger

PY - 2018/9/10

Y1 - 2018/9/10

N2 - Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs). It was shown that DCNN s are stable with respect to deformations of bandlimited input functions. In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces. Further, we show a weak form of deformation stability for the whole input space L2(Rd). The basic components of DCNNs are semi-discrete frames. For practical applications, a concrete choice is necessary. Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.

AB - Our work is based on a recently introduced mathematical theory of deep convolutional neural networks (DCNNs). It was shown that DCNN s are stable with respect to deformations of bandlimited input functions. In the present paper, we generalize this result: We prove deformation stability on Sobolev spaces. Further, we show a weak form of deformation stability for the whole input space L2(Rd). The basic components of DCNNs are semi-discrete frames. For practical applications, a concrete choice is necessary. Therefore, we conclude our work by suggesting a construction method for semi-discrete frames based on bounded uniform partitions of unity (BUPUs) and give a specific example that uses B-splines.

KW - Admissible semi -discrete frame

KW - Bounded uniform partition of unity

KW - Deep convolutional neural networks

KW - Deformation stability

KW - Sobolev space

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BT - 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings

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Koller M, Grobmann J, Monich U, Boche H. Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces. In 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 - Proceedings. Institute of Electrical and Electronics Engineers Inc. 2018. p. 6872-6876. 8462158. (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings). doi: 10.1109/ICASSP.2018.8462158

Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces

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