# Learning Invariant Representations for Equivariant Neural Networks Using Orthogonal Moments

arXiv: https://arxiv.org/abs/2209.10944 · HTML: https://mlsanity.com/paper/2209.10944/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 27 references that are in the map.
- First posted on arXiv 22 Sep 2022.

## Builds on (most cited first, 10 of 10)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Equivariant Convolutional Networks: Application to Histopathology Image Analysis](https://mlsanity.com/paper/2002.08725/) (2020)
- [HexaConv](https://mlsanity.com/paper/1803.02108/) (2018)

## Built on by (most cited first, 3 of 3)

- [H-NeXt: The next step towards roto-translation invariant networks](https://mlsanity.com/paper/2311.01111/) (2023)
- [Few-shot Shape Recognition by Learning Deep Shape-aware Features](https://mlsanity.com/paper/2312.01315/) (2023)
- [Entropy Transformer Networks: A Learning Approach via Tangent Bundle Data Manifold](https://mlsanity.com/paper/2307.12517/) (2023)

## Abstract (arXiv)

The convolutional layers of standard convolutional neural networks (CNNs) are equivariant to translation. However, the convolution and fully-connected layers are not equivariant or invariant to other affine geometric transformations. Recently, a new class of CNNs is proposed in which the conventional layers of CNNs are replaced with equivariant convolution, pooling, and batch-normalization layers. The final classification layer in equivariant neural networks is invariant to different affine geometric transformations such as rotation, reflection and translation, and the scalar value is obtained by either eliminating the spatial dimensions of filter responses using convolution and down-sampling throughout the network or average is taken over the filter responses. In this work, we propose to integrate the orthogonal moments which gives the high-order statistics of the function as an effective means for encoding global invariance with respect to rotation, reflection and translation in fully-connected layers. As a result, the intermediate layers of the network become equivariant while the classification layer becomes invariant. The most widely used Zernike, pseudo-Zernike and orthogonal Fourier-Mellin moments are considered for this purpose. The effectiveness of the proposed work is evaluated by integrating the invariant transition and fully-connected layer in the architecture of group-equivariant CNNs (G-CNNs) on rotated MNIST and CIFAR10 datasets.