Deep Convolutional Neural Networks on Cartoon Functions
This work addresses a theoretical limitation for researchers in signal processing and machine learning, providing a more applicable stability result for non-band-limited signals.
The paper tackles the problem of deformation stability in deep convolutional neural networks for signals with sharp discontinuities, establishing stability bounds for cartoon functions.
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.