Group Equivariant Convolutional Neural Networks
Symmetries are the main source of unifying principles in maths and physics."
~ Peter Bürgisser
Implicit representation learning doesn't leverage structure, they overfit a particular training task.
As shown by T Cohen. et al (2017), we can learn representations that have the structure of a linear G-space for some chosen group G. The structure can be preserved from input space to the representation space by making them equivariant:
$$
f(T_gx) = T' f(x)
$$
What about invariance?
Equivariance: Output transforms predictably with input
$$
f(T(x)) = T'(f(x))
$$
Invariance: Output stays unchanged despite input transformation
$$
f(T(x)) = f(x)
$$
Key insight: Invariance is equivariance where $T' = I$ (identity)
When to use each:
- Equivariance: When output should reflect input transformations (e.g., feature maps in CNNs)
- Invariance: When output should ignore transformations (e.g., classification - rotated cat is still a cat)
Structure transfer:
- Equivariance preserves and inherits transformation structure
- Invariance discards transformation information while preserving semantic content
- Both enable sample efficiency by learning patterns that generalize across transformations.