# A General Framework for Robust G-Invariance in G-Equivariant Networks

arXiv: https://arxiv.org/abs/2310.18564 · HTML: https://mlsanity.com/paper/2310.18564/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 58 references that are in the map.
- First posted on arXiv 28 Oct 2023.

## Builds on (most cited first, 25 of 25)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spatial Pyramid Pooling in Deep Convolutional Networks for Visual Recognition](https://mlsanity.com/paper/1406.4729/) (2014)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Approximating CNNs with Bag-of-local-Features models works surprisingly well on ImageNet](https://mlsanity.com/paper/1904.00760/) (2019)
- [Bilinear CNNs for Fine-grained Visual Recognition](https://mlsanity.com/paper/1504.07889/) (2015)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [Excessive Invariance Causes Adversarial Vulnerability](https://mlsanity.com/paper/1811.00401/) (2018)
- [Global Second-order Pooling Convolutional Networks](https://mlsanity.com/paper/1811.12006/) (2018)
- [Stochastic Pooling for Regularization of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1301.3557/) (2013)
- [Fractional Max-Pooling](https://mlsanity.com/paper/1412.6071/) (2014)
- [Learned-Norm Pooling for Deep Feedforward and Recurrent Neural Networks](https://mlsanity.com/paper/1311.1780/) (2013)
- [Performance of a Geometric Deep Learning Pipeline for HL-LHC Particle Tracking](https://mlsanity.com/paper/2103.06995/) (2021)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [S3Pool: Pooling with Stochastic Spatial Sampling](https://mlsanity.com/paper/1611.05138/) (2016)
- [Signal Recovery from Pooling Representations](https://mlsanity.com/paper/1311.4025/) (2013)
- [Max-Pooling Dropout for Regularization of Convolutional Neural Networks](https://mlsanity.com/paper/1512.01400/) (2015)
- [Comparison of Methods Generalizing Max- and Average-Pooling](https://mlsanity.com/paper/2103.01746/) (2021)
- [Alpha-Integration Pooling for Convolutional Neural Networks](https://mlsanity.com/paper/1811.03436/) (2018)
- [Multipartite Pooling for Deep Convolutional Neural Networks](https://mlsanity.com/paper/1710.07435/) (2017)
- [Ordinal Pooling Networks: For Preserving Information over Shrinking Feature Maps](https://mlsanity.com/paper/1804.02702/) (2018)
- [Maximal function pooling with applications](https://mlsanity.com/paper/2103.01292/) (2021)

## Built on by (most cited first, 5 of 5)

- [The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks](https://mlsanity.com/paper/2407.07655/) (2024)
- [Quick ViTs: Speeding up Vision Transformers through Equivariance](https://mlsanity.com/paper/2505.15441/) (2025)
- [Selective Disk Bispectrum: A Complete and Rotation Invariant Image Descriptor](https://mlsanity.com/paper/2511.19706/) (2025)
- [Rotation-Invariant Spherical Watermarking via Third-Order SO(3) Representation Coupling](https://mlsanity.com/paper/2605.26702/) (2026)
- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)

## Abstract (arXiv)

We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks (G-CNNs), which we call the G-triple-correlation (G-TC) layer. The approach leverages the theory of the triple-correlation on groups, which is the unique, lowest-degree polynomial invariant map that is also complete. Many commonly used invariant maps--such as the max--are incomplete: they remove both group and signal structure. A complete invariant, by contrast, removes only the variation due to the actions of the group, while preserving all information about the structure of the signal. The completeness of the triple correlation endows the G-TC layer with strong robustness, which can be observed in its resistance to invariance-based adversarial attacks. In addition, we observe that it yields measurable improvements in classification accuracy over standard Max G-Pooling in G-CNN architectures. We provide a general and efficient implementation of the method for any discretized group, which requires only a table defining the group's product structure. We demonstrate the benefits of this method for G-CNNs defined on both commutative and non-commutative groups--SO(2), O(2), SO(3), and O(3) (discretized as the cyclic C8, dihedral D16, chiral octahedral O and full octahedral Oh groups)--acting on ℝ2 and ℝ3 on both G-MNIST and G-ModelNet10 datasets.