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On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

Risi Kondor, Shubhendu Trivedi

TL;DR

This paper develops a rigorous framework linking neural network equivariance to generalized group convolutions for compact groups. By formulating convolutions on groups and their quotient spaces and applying noncommutative harmonic analysis, it proves that, under natural constraints, equivariant networks are exactly those built from generalized convolutions. The authors illustrate the theory with concrete examples, including rotation- and spherical-symmetry networks and graph-based message passing, showing how Fourier-domain perspectives govern filter design. The work provides a unifying language and design principles for symmetry-aware architectures across non-Euclidean domains.

Abstract

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.

On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

TL;DR

This paper develops a rigorous framework linking neural network equivariance to generalized group convolutions for compact groups. By formulating convolutions on groups and their quotient spaces and applying noncommutative harmonic analysis, it proves that, under natural constraints, equivariant networks are exactly those built from generalized convolutions. The authors illustrate the theory with concrete examples, including rotation- and spherical-symmetry networks and graph-based message passing, showing how Fourier-domain perspectives govern filter design. The work provides a unifying language and design principles for symmetry-aware architectures across non-Euclidean domains.

Abstract

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.

Paper Structure

This paper contains 30 sections, 14 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Equivariance in neural networks
  4. Group actions
  5. Equivariance
  6. Convolution on groups and quotient spaces
  7. Convolution on quotient spaces
  8. Relationship to Fourier analysis
  9. Main result: the connection between convolution and equivariance
  10. Examples of algebraic convolution in neural networks
  11. Rotation equivariant networks
  12. Spherical CNNs
  13. Message passing neural networks
  14. Conclusions
  15. Background from group and representation theory
  16. ...and 15 more sections

Key Result

Theorem 1

A feed forward neural network $\mathcal{N}$ is equivariant to the action of a compact group $G$ on its inputs if and only if each layer of $\mathcal{N}$ implements a generalized form of convolution derived from (eq: convo0).

Theorems & Definitions (22)

  • Theorem 1
  • Definition 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Proposition 2: Convolution theorem on groups
  • Definition 5
  • Theorem 1
  • ...and 12 more