# Affine Invariance in Continuous-Domain Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2311.09245 · HTML: https://mlsanity.com/paper/2311.09245/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 34 references that are in the map.
- First posted on arXiv 13 Nov 2023.

## Builds on (most cited first, 18 of 18)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [An Introduction to Convolutional Neural Networks](https://mlsanity.com/paper/1511.08458/) (2015)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [The Computational Limits of Deep Learning](https://mlsanity.com/paper/2007.05558/) (2020)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [Oriented Response Networks](https://mlsanity.com/paper/1701.01833/) (2017)
- [CubeNet: Equivariance to 3D Rotation and Translation](https://mlsanity.com/paper/1804.04458/) (2018)
- [Deep Roto-Translation Scattering for Object Classification](https://mlsanity.com/paper/1412.8659/) (2014)
- [3D G-CNNs for Pulmonary Nodule Detection](https://mlsanity.com/paper/1804.04656/) (2018)
- [PDE-based Group Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2001.09046/) (2020)
- [Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups](https://mlsanity.com/paper/2110.13059/) (2021)
- [An optimized system to solve text-based CAPTCHA](https://mlsanity.com/paper/1806.07202/) (2018)

## Abstract (arXiv)

The notion of group invariance helps neural networks in recognizing patterns and features under geometric transformations. Group convolutional neural networks enhance traditional convolutional neural networks by incorporating group-based geometric structures into their design. This research studies affine invariance on continuous-domain convolutional neural networks. Despite other research considering isometric invariance or similarity invariance, we focus on the full structure of affine transforms generated by the group of all invertible 2 × 2 real matrices (generalized linear group GL2(ℝ)). We introduce a new criterion to assess the invariance of two signals under affine transformations. The input image is embedded into the affine Lie group G2 = ℝ2 ltimes GL2(ℝ) to facilitate group convolution operations that respect affine invariance. Then, we analyze the convolution of embedded signals over G2. In sum, our research could eventually extend the scope of geometrical transformations that usual deep-learning pipelines can handle.