Deep convolutional neural networks on cartoon functions

Authors

Philipp Grohs, Thomas Wiatowski, and Helmut Bölcskei

Reference

Proc. of IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, pp. 1163-1167, July 2016.

Abstract

Wiatowski and Bölcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific convolution kernels and non-linearities. While the translation invariance result applies to square-integrable functions, the deformation stability bound holds for band-limited functions only. Many signals of practical relevance (such as natural images) exhibit, however, sharp and curved discontinuities and are, hence, not band-limited. The main contribution of this paper is a deformation stability result that takes these structural properties into account. Specifically, we establish deformation stability bounds for the class of cartoon functions introduced by Donoho, 2001.

Keywords

Deep neural networks, deformation stability, cartoon functions

Comments

This is a slightly updated version of the paper published in the ISIT proceedings. Specifically, we corrected errors in the arguments on the volume of tubes. Note that this correction does not affect the main statements of the paper.

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