Deep Convolutional Neural Networks on Cartoon Functions

TL;DR

This paper extends the theoretical understanding of deep convolutional neural networks by establishing deformation stability bounds specifically for cartoon functions, which better model natural images with sharp discontinuities.

Contribution

It provides a new deformation stability result tailored for cartoon functions, addressing limitations of previous bounds that required band-limited signals.

Findings

Deformation stability bounds for cartoon functions are established.

The results apply to natural images with sharp discontinuities.

Theoretical guarantees are extended beyond band-limited functions.

Abstract

Wiatowski and B\"olcskei, 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.