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Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning

Ramzan Basheer, Deepak Mishra

TL;DR

The paper surveys symmetry group equivariant convolution frameworks for representation learning on graphs and manifolds, classifying them into regular, steerable, and PDE-based families. It explains how group theory underpins equivariance and discusses how these methods extend to Euclidean, spherical, and manifold domains through lifting, intertwiners, and gauge concepts. It covers geometric priors, datasets, and applications in computer vision and dynamic physical simulations, and outlines challenges such as data scarcity and generalization, along with promising directions like unsupervised geometric learning and hypercomplex representations. By organizing the landscape and clarifying methodological trade-offs, the work provides a comprehensive reference for researchers aiming to develop principled, symmetry-aware representations in non-Euclidean domains.

Abstract

Euclidean deep learning is often inadequate for addressing real-world signals where the representation space is irregular and curved with complex topologies. Interpreting the geometric properties of such feature spaces has become paramount in obtaining robust and compact feature representations that remain unaffected by nontrivial geometric transformations, which vanilla CNNs cannot effectively handle. Recognizing rotation, translation, permutation, or scale symmetries can lead to equivariance properties in the learned representations. This has led to notable advancements in computer vision and machine learning tasks under the framework of geometric deep learning, as compared to their invariant counterparts. In this report, we emphasize the importance of symmetry group equivariant deep learning models and their realization of convolution-like operations on graphs, 3D shapes, and non-Euclidean spaces by leveraging group theory and symmetry. We categorize them as regular, steerable, and PDE-based convolutions and thoroughly examine the inherent symmetries of their input spaces and ensuing representations. We also outline the mathematical link between group convolutions or message aggregation operations and the concept of equivariance. The report also highlights various datasets, their application scopes, limitations, and insightful observations on future directions to serve as a valuable reference and stimulate further research in this emerging discipline.

Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning

TL;DR

The paper surveys symmetry group equivariant convolution frameworks for representation learning on graphs and manifolds, classifying them into regular, steerable, and PDE-based families. It explains how group theory underpins equivariance and discusses how these methods extend to Euclidean, spherical, and manifold domains through lifting, intertwiners, and gauge concepts. It covers geometric priors, datasets, and applications in computer vision and dynamic physical simulations, and outlines challenges such as data scarcity and generalization, along with promising directions like unsupervised geometric learning and hypercomplex representations. By organizing the landscape and clarifying methodological trade-offs, the work provides a comprehensive reference for researchers aiming to develop principled, symmetry-aware representations in non-Euclidean domains.

Abstract

Euclidean deep learning is often inadequate for addressing real-world signals where the representation space is irregular and curved with complex topologies. Interpreting the geometric properties of such feature spaces has become paramount in obtaining robust and compact feature representations that remain unaffected by nontrivial geometric transformations, which vanilla CNNs cannot effectively handle. Recognizing rotation, translation, permutation, or scale symmetries can lead to equivariance properties in the learned representations. This has led to notable advancements in computer vision and machine learning tasks under the framework of geometric deep learning, as compared to their invariant counterparts. In this report, we emphasize the importance of symmetry group equivariant deep learning models and their realization of convolution-like operations on graphs, 3D shapes, and non-Euclidean spaces by leveraging group theory and symmetry. We categorize them as regular, steerable, and PDE-based convolutions and thoroughly examine the inherent symmetries of their input spaces and ensuing representations. We also outline the mathematical link between group convolutions or message aggregation operations and the concept of equivariance. The report also highlights various datasets, their application scopes, limitations, and insightful observations on future directions to serve as a valuable reference and stimulate further research in this emerging discipline.
Paper Structure (38 sections, 34 equations, 4 figures, 3 tables)

This paper contains 38 sections, 34 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Geometric Deep Learning
  4. Equivariance
  5. Steerable Representations
  6. Domains of Equivariant Learning
  7. Graphs
  8. Manifolds
  9. Group Equivariant Convolutions
  10. Regular Group Convolutions
  11. Discrete Setting
  12. Continuous Setting
  13. Steerable Group Convolutions
  14. PDE-based Group Convolutions
  15. Geometric Priors
  16. ...and 23 more sections

Figures (4)

  • Figure 1: A taxonomy of symmetry group equivariant representation learning.
  • Figure 2: A block diagram showing equivariant action on a network layer.
  • Figure 3: An example to show group equivariant convolution. a) Convolution output of inputs under no group action. b) Convolution output with respect to some group actions on input feature space. c) Convolution output with respect to group actions on output feature space
  • Figure 4: Spherical convolution