# Learning Stable Group Invariant Representations with Convolutional Networks

arXiv: https://arxiv.org/abs/1301.3537 · HTML: https://mlsanity.com/paper/1301.3537/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2013.

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

- [Johnson-Lindenstrauss Lemma, Linear and Nonlinear Random Projections, Random Fourier Features, and Random Kitchen Sinks: Tutorial and Survey](https://mlsanity.com/paper/2108.04172/) (2021)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [Quantised Transforming Auto-Encoders: Achieving Equivariance to Arbitrary Transformations in Deep Networks](https://mlsanity.com/paper/2111.12873/) (2021)
- [Towards the One Learning Algorithm Hypothesis: A System-theoretic Approach](https://mlsanity.com/paper/2112.02256/) (2021)

## Abstract (arXiv)

Transformation groups, such as translations or rotations, effectively express part of the variability observed in many recognition problems. The group structure enables the construction of invariant signal representations with appealing mathematical properties, where convolutions, together with pooling operators, bring stability to additive and geometric perturbations of the input. Whereas physical transformation groups are ubiquitous in image and audio applications, they do not account for all the variability of complex signal classes. We show that the invariance properties built by deep convolutional networks can be cast as a form of stable group invariance. The network wiring architecture determines the invariance group, while the trainable filter coefficients characterize the group action. We give explanatory examples which illustrate how the network architecture controls the resulting invariance group. We also explore the principle by which additional convolutional layers induce a group factorization enabling more abstract, powerful invariant representations.