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Lie Groups & Equivariant Networks

CNNs are equivariant to translations—this explains why they work for images

Concept 73 of 100TheoryPhase 10
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#73Lie GroupsTheory
key equation
f(g⋅x)=g⋅f(x)f(g \cdot x) = g \cdot f(x)
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Concept 73 of 100Lie GroupsTheory / Phase 10: Mathematical foundations & information geometry
Current question

Equivariance = "the output transforms the same way as the input"

f(g⋅x)=g⋅f(x)f(g \cdot x) = g \cdot f(x)
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Why It Matters for Modern Models

  • CNNs are equivariant to translations—this explains why they work for images
  • Symmetry constraints dramatically reduce parameter count and improve generalization
  • Molecular/protein prediction requires 3D rotation equivariance (SE(3))

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • Equivariance = "the output transforms the same way as the input"
  • Translation equivariance (CNN) is the simplest case; rotation, scale are harder
  • The Lie algebra captures "infinitesimal" symmetries—rotation by tiny angles

Visualization Status

Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and cited sources. Come back here for the full walkthrough.

Key Equation
f(g⋅x)=g⋅f(x)f(g \cdot x) = g \cdot f(x)

A Lie group GG is a smooth manifold with group structure.

Equivariance: f(g⋅x)=g⋅f(x)f(g \cdot x) = g \cdot f(x) for all g∈Gg \in G

Group convolution (generalizes 2D convolution):

[f∗ψ](g)=∫Gf(h)ψ(g−1h)dh[f * \psi](g) = \int_G f(h) \psi(g^{-1}h) dh

Lie algebra g\mathfrak{g}: infinitesimal generators at identity:

exp⁡:g→G\exp: \mathfrak{g} \to G

Rotation group SO(3)SO(3): 3D rotations, Lie algebra is skew-symmetric matrices.

Canonical Papers

A General Theory of Equivariant CNNs on Homogeneous Spaces

Cohen et al.2019NeurIPS
Read paper →

Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

Bronstein et al.2021arXiv
Read paper →

Connections

Prerequisites

∀Theory🖼️Multimodal VLP

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