Ask what should change when the equation is manipulated, then let the visualization test that expectation.
Foundation Lab
Lie Groups & Equivariant Networks
CNNs are equivariant to translations—this explains why they work for images
#73Lie GroupsTheory
key equation
f(g \cdot x) = g \cdot f(x)Selected Foundation Object
Keep the equation fixed; move through the evidence.
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 \cdot x) = g \cdot f(x)Use the runnable panel, the key equation, and canonical papers as separate forms of evidence for the same object.
The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.
Use prerequisites, dependents, and semantic links to repair the next gap without leaving the object behind.
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
Interactive Visualization
Core Math (Optional Deep Dive)
If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.
Key Equation
A Lie group is a smooth manifold with group structure.
Equivariance: for all
Group convolution (generalizes 2D convolution):
Lie algebra : infinitesimal generators at identity:
Rotation group : 3D rotations, Lie algebra is skew-symmetric matrices.