# Stability Analysis of Equivariant Convolutional Representations Through The Lens of Equivariant Multi-layered CKNs

arXiv: https://arxiv.org/abs/2408.04277 · HTML: https://mlsanity.com/paper/2408.04277/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 41 references that are in the map.
- First posted on arXiv 8 Aug 2024.

## Builds on (most cited first, 27 of 27)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [Convolutional Kernel Networks](https://mlsanity.com/paper/1406.3332/) (2014)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [Why Are Convolutional Nets More Sample-Efficient than Fully-Connected Nets?](https://mlsanity.com/paper/2010.08515/) (2020)
- [Approximation and Learning with Deep Convolutional Models: a Kernel Perspective](https://mlsanity.com/paper/2102.10032/) (2021)
- [End-to-End Kernel Learning with Supervised Convolutional Kernel Networks](https://mlsanity.com/paper/1605.06265/) (2016)
- [DCFNet: Deep Neural Network with Decomposed Convolutional Filters](https://mlsanity.com/paper/1802.04145/) (2018)
- [A PAC-Bayesian Generalization Bound for Equivariant Networks](https://mlsanity.com/paper/2210.13150/) (2022)
- [Local Group Invariant Representations via Orbit Embeddings](https://mlsanity.com/paper/1612.01988/) (2016)
- [Deep Convolutional Networks are Hierarchical Kernel Machines](https://mlsanity.com/paper/1508.01084/) (2015)
- [Convexified Convolutional Neural Networks](https://mlsanity.com/paper/1609.01000/) (2016)
- [Deformation Robust Roto-Scale-Translation Equivariant CNNs](https://mlsanity.com/paper/2111.10978/) (2021)
- [Provable Adversarial Robustness for Group Equivariant Tasks: Graphs, Point Clouds, Molecules, and More](https://mlsanity.com/paper/2312.02708/) (2023)

## Abstract (arXiv)

In this paper we construct and theoretically analyse group equivariant convolutional kernel networks (CKNs) which are useful in understanding the geometry of (equivariant) CNNs through the lens of reproducing kernel Hilbert spaces (RKHSs). We then proceed to study the stability analysis of such equiv-CKNs under the action of diffeomorphism and draw a connection with equiv-CNNs, where the goal is to analyse the geometry of inductive biases of equiv-CNNs through the lens of reproducing kernel Hilbert spaces (RKHSs). Traditional deep learning architectures, including CNNs, trained with sophisticated optimization algorithms is vulnerable to perturbations, including `adversarial examples'. Understanding the RKHS norm of such models through CKNs is useful in designing the appropriate architecture and can be useful in designing robust equivariant representation learning models.