A Brief History & Overview

This video provides a brief context and historical overview of the development of Group Equivariant Deep Learning (GEDL).

1. The Seminal Paper (2016)

The field was notably kickstarted by Cohen & Welling with their ICML 2016 paper "Group Equivariant Convolutional Networks".

• Key Contribution: Formalized the G-CNN framework for discrete groups (e.g., 90-degree rotations, reflections).
• Method: Rotating kernels by 90 degrees can be implemented as exact permutations of pixels, avoiding interpolation artifacts.

2. Moving to 3D and Continuous Groups

Following the 2016 paper, the field expanded rapidly:

• 3D G-CNNs: Extended the concept to 3D voxel grids (e.g., cube symmetries).
• Hexagonal Grids: Used hexagonal pixels to allow for 6 discrete rotations (multiples of 60 degrees) instead of just 4.
• Continuous Rotations: Researchers began tackling the continuous rotation group $SO(2)$. This led to two main approaches:
  • Interpolation-based: Rotating the kernel and using bilinear interpolation (what we discussed in Lecture 1.3).
  • Steerable G-CNNs: Using a basis of functions that rotate analytically (e.g., circular harmonics). This avoids interpolation artifacts and handles the continuous group exactly. We cover this in Module 2.

3. Lie Groups & Manifolds

The framework was further generalized to:

• Lie Groups: Handling groups with continuous parameters like scale and affine transformations using Lie algebra tools (Exponential map, Log map).
• Manifolds (Gauge Equivariance): The paper "Coordinate Independent Convolutional Networks" (Weiler et al.) laid the groundwork for defining convolutions on general manifolds using local gauge frames. This is crucial for data on meshes or curved surfaces (e.g., earth climate data, brain signals).

4. The Modern Landscape

Today, equivariance is a standard inductive bias in deep learning, especially for scientific applications.