# Understanding Deep Convolutional Networks

arXiv: https://arxiv.org/abs/1601.04920 · HTML: https://mlsanity.com/paper/1601.04920/

## Key facts

- Cited by 76 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jan 2016.

## Built on by (most cited first, 30 of 76)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Data-driven emergence of convolutional structure in neural networks](https://mlsanity.com/paper/2202.00565/) (2022)
- [Physics-Informed Computer Vision: A Review and Perspectives](https://mlsanity.com/paper/2305.18035/) (2023)
- [Learning sparse features can lead to overfitting in neural networks](https://mlsanity.com/paper/2206.12314/) (2022)
- [Theory of Deep Convolutional Neural Networks III: Approximating Radial Functions](https://mlsanity.com/paper/2107.00896/) (2021)
- [Low-Fidelity End-to-End Video Encoder Pre-training for Temporal Action Localization](https://mlsanity.com/paper/2103.15233/) (2021)
- [Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs](https://mlsanity.com/paper/2102.02828/) (2021)
- [Automatic Data Augmentation via Invariance-Constrained Learning](https://mlsanity.com/paper/2209.15031/) (2022)
- [Scaling Properties of Deep Residual Networks](https://mlsanity.com/paper/2105.12245/) (2021)
- [Parametric Scattering Networks](https://mlsanity.com/paper/2107.09539/) (2021)
- [Relative stability toward diffeomorphisms indicates performance in deep nets](https://mlsanity.com/paper/2105.02468/) (2021)
- [High-dimensional learning of narrow neural networks](https://mlsanity.com/paper/2409.13904/) (2024)
- [On the Regularity of Attention](https://mlsanity.com/paper/2102.05628/) (2021)
- [Wavelet Conditional Renormalization Group](https://mlsanity.com/paper/2207.04941/) (2022)
- [Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2303.09803/) (2023)
- [Tailoring Self-Supervision for Supervised Learning](https://mlsanity.com/paper/2207.10023/) (2022)
- [How Deep Networks Learn Sparse and Hierarchical Data: the Sparse Random Hierarchy Model](https://mlsanity.com/paper/2404.10727/) (2024)
- [Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis](https://mlsanity.com/paper/2207.10939/) (2022)
- [Normalization effects on deep neural networks](https://mlsanity.com/paper/2209.01018/) (2022)
- [Sparsity in long-time control of neural ODEs](https://mlsanity.com/paper/2102.13566/) (2021)
- [Convolutional Neural Networks for Spherical Signal Processing via Spherical Haar Tight Framelets](https://mlsanity.com/paper/2201.07890/) (2022)
- [Decision boundaries and convex hulls in the feature space that deep learning functions learn from images](https://mlsanity.com/paper/2202.04052/) (2022)
- [Equilibrium Aggregation: Encoding Sets via Optimization](https://mlsanity.com/paper/2202.12795/) (2022)
- [Generalized Permutants and Graph GENEOs](https://mlsanity.com/paper/2206.14798/) (2022)
- [How deep convolutional neural networks lose spatial information with training](https://mlsanity.com/paper/2210.01506/) (2022)
- [Fairness in Machine Learning meets with Equity in Healthcare](https://mlsanity.com/paper/2305.07041/) (2023)
- [GENEOnet: A new machine learning paradigm based on Group Equivariant Non-Expansive Operators. An application to protein pocket detection](https://mlsanity.com/paper/2202.00451/) (2022)
- [Riesz feature representation: scale equivariant scattering network for classification tasks](https://mlsanity.com/paper/2307.08467/) (2023)
- [Spectral Norm of Convolutional Layers with Circular and Zero Paddings](https://mlsanity.com/paper/2402.00240/) (2024)
- [Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators](https://mlsanity.com/paper/2405.05095/) (2024)

## Abstract (arXiv)

Deep convolutional networks provide state of the art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and non-linearities. A mathematical framework is introduced to analyze their properties. Computations of invariants involve multiscale contractions, the linearization of hierarchical symmetries, and sparse separations. Applications are discussed.