# Deep Convolutional Neural Networks on Cartoon Functions

arXiv: https://arxiv.org/abs/1605.00031 · HTML: https://mlsanity.com/paper/1605.00031/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2016.

## Built on by (most cited first, 4 of 4)

- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [Manifold Filter-Combine Networks](https://mlsanity.com/paper/2307.04056/) (2023)
- [Generalized moduli of continuity under irregular or random deformations via multiscale analysis](https://mlsanity.com/paper/2104.11977/) (2021)
- [Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily Slow](https://mlsanity.com/paper/2406.05121/) (2024)

## Abstract (arXiv)

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.