Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces
Michael Koller, Johannes Grobmann, Ullrich J. Mönich, Holger Boche
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 L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (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.
BibTeX
@inproceedings{icassp2018_deformationstabi,
title = {Deformation Stability of Deep Convolutional Neural Networks on Sobolev Spaces},
author = {Michael Koller and Johannes Grobmann and Ullrich J. Mönich and Holger Boche},
booktitle = {ICASSP 2018},
year = {2018}
}