# Deep Convolutional Neural Networks Based on Semi-Discrete Frames

arXiv: https://arxiv.org/abs/1504.05487 · HTML: https://mlsanity.com/paper/1504.05487/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2015.

## Built on by (most cited first, 6 of 6)

- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [Stability of the scattering transform for deformations with minimal regularity](https://mlsanity.com/paper/2205.11142/) (2022)
- [Generalized moduli of continuity under irregular or random deformations via multiscale analysis](https://mlsanity.com/paper/2104.11977/) (2021)
- [CoShNet: A Hybrid Complex Valued Neural Network using Shearlets](https://mlsanity.com/paper/2208.06882/) (2022)
- [VDW-GNNs: Vector diffusion wavelets for geometric graph neural networks](https://mlsanity.com/paper/2510.01022/) (2025)
- [Transversal Pooling Neural Networks](https://mlsanity.com/paper/2609.36237/) (2026)

## Abstract (arXiv)

Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on identical semi-discrete wavelet frames in each network layer, and proved translation-invariance as well as deformation stability of the resulting feature extractor. The purpose of this paper is to develop Mallat's theory further by allowing for different and, most importantly, general semi-discrete frames (such as, e.g., Gabor frames, wavelets, curvelets, shearlets, ridgelets) in distinct network layers. This allows to extract wider classes of features than point singularities resolved by the wavelet transform. Our generalized feature extractor is proven to be translation-invariant, and we develop deformation stability results for a larger class of deformations than those considered by Mallat. For Mallat's wavelet-based feature extractor, we get rid of a number of technical conditions. The mathematical engine behind our results is continuous frame theory, which allows us to completely detach the invariance and deformation stability proofs from the particular algebraic structure of the underlying frames.