# Energy Propagation in Scattering Convolution Networks Can Be Arbitrarily Slow

arXiv: https://arxiv.org/abs/2406.05121 · HTML: https://mlsanity.com/paper/2406.05121/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 7 Jun 2024.

## Builds on (most cited first, 14 of 14)

- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Geometric Scattering for Graph Data Analysis](https://mlsanity.com/paper/1810.03068/) (2018)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Geometric Scattering on Measure Spaces](https://mlsanity.com/paper/2208.08561/) (2022)
- [Solid Harmonic Wavelet Scattering for Predictions of Molecule Properties](https://mlsanity.com/paper/1805.00571/) (2018)
- [Joint Time-Frequency Scattering for Audio Classification](https://mlsanity.com/paper/1512.02125/) (2015)
- [Wavelet Scattering Regression of Quantum Chemical Energies](https://mlsanity.com/paper/1605.04654/) (2016)
- [Energy Propagation in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1704.03636/) (2017)
- [Deep Convolutional Neural Networks on Cartoon Functions](https://mlsanity.com/paper/1605.00031/) (2016)
- [Wavelet Scattering Networks for Atomistic Systems with Extrapolation of Material Properties](https://mlsanity.com/paper/2006.01247/) (2020)
- [Topology Reduction in Deep Convolutional Feature Extraction Networks](https://mlsanity.com/paper/1707.02711/) (2017)
- [Wavelets Beat Monkeys at Adversarial Robustness](https://mlsanity.com/paper/2304.09403/) (2023)
- [Discrete Deep Feature Extraction: A Theory and New Architectures](https://mlsanity.com/paper/1605.08283/) (2016)

## Built on by (most cited first, 1 of 1)

- [Digital implementations of deep feature extractors are intrinsically informative](https://mlsanity.com/paper/2502.15004/) (2025)

## Abstract (arXiv)

We analyze energy decay for deep convolutional neural networks employed as feature extractors, including Mallat's wavelet scattering transform. For time-frequency scattering transforms based on Gabor filters, previous work has established that energy decay is exponential for arbitrary square-integrable input signals. In contrast, our main results allow proving that this is false for wavelet scattering in arbitrary dimensions. Specifically, we show that the energy decay of wavelet and wavelet-like scattering transforms acting on generic square-integrable signals can be arbitrarily slow. Importantly, this slow decay behavior holds for dense subsets of L2(ℝd), indicating that rapid energy decay is generally an unstable property of signals. We complement these findings with positive results that allow us to infer fast (up to exponential) energy decay for generalized Sobolev spaces tailored to the frequency localization of the underlying filter bank. Both negative and positive results highlight that energy decay in scattering networks critically depends on the interplay between the respective frequency localizations of both the signal and the filters used.