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

TL;DR

This paper develops and analyzes group equivariant convolutional kernel networks (CKNs) to understand the stability and geometry of equivariant CNNs using reproducing kernel Hilbert spaces, with implications for robustness against perturbations.

Contribution

It introduces a theoretical framework for equivariant CKNs, connecting their stability properties to the geometry of equivariant CNNs via RKHS analysis.

Findings

Equivariant CKNs provide insights into the stability of equivariant CNNs.

RKHS norms help in understanding model robustness.

The analysis suggests ways to design more robust equivariant architectures.

Abstract

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.