Interactive Explainer

Equivariant Networks, Live

A neural network is equivariant to a transformation when rotating the input rotates the output the same way. For satellite imagery, molecules, point clouds, and any system without a privileged orientation, this is what you want and most vanilla networks don't deliver it.

~16 min Inductive Bias · Symmetry · CNN

Step 1

Equivariance vs invariance

$f$ is equivariant to a transformation $T_g$ if applying $T_g$ to the input has the same effect as applying some (possibly different) transformation to the output. Invariance is the special case where the output doesn't change at all.

Translation equivariance. Move an object sideways — feature maps shift the same way. Standard CNN convolution gives you this for free (modulo padding).
Rotation equivariance. Rotate the input by 90° — feature maps rotate by 90° too. Standard CNNs don't have this; you'd need C4 group convolutions (Cohen & Welling, 2016).
Scale equivariance. Zoom in — features scale. Mostly approximated via pyramids; true equivariant architectures exist but are rare.
SE(3) equivariance. 3-D rigid-body symmetries. Critical for molecules / point clouds (E(3)NN, SE(3)-Transformer).
Permutation equivariance. Sets and graphs. Self-attention is permutation-equivariant; GNN aggregations are too.

Step 2

Why a vanilla CNN can't rotate

A standard 3×3 convolution filter is a fixed pattern in pixel space. Rotating the input does not rotate the filter; the response thrashes. Three classical fixes:

Data augmentation. Train with random rotations. Cheap; nearly works for classification; the response is approximately invariant but not equivariant, and the model still has to learn each rotation redundantly.
Group convolutions (G-CNN). Apply $k$ rotated copies of the same filter at each layer; the feature map has an extra "rotation" axis. Cohen & Welling 2016; provably equivariant.
Steerable CNNs. Use filters that transform continuously under rotation (linear combinations of circular harmonics). Equivariant to the continuous $\mathrm{SO}(2)$ group, not just discrete.

Step 3

Live: rotate the input, watch three architectures respond

A synthetic airplane-silhouette input is rotated by θ. Three tiny networks emit a feature vector. Plot the L2 distance between the feature at θ=0 and the feature at the current θ. A perfectly invariant network would output a flat line at zero; a perfectly equivariant one would show a clean orbit shape. The vanilla CNN responds erratically.

rotation θ 0°

input at θ

vanilla CNN response

augmented CNN response

C4 group conv response

Feature distance from θ=0, swept over θ

Step 4

Where to use what

Satellite / aerial imagery. Sentinel and Landsat tiles have no canonical "up"; aerial / drone footage rotates with the platform. C4 / C8 group convolutions on the backbone, or rotation augmentation + test-time average-of-rotations.
Molecular & atomic systems. Energies are invariant to rigid rotations; forces are equivariant. Use E(3)NN, SE(3)-Transformer, MACE for molecular ML; AlphaFold-2's invariant point attention is in this family.
Point clouds. 3-D LiDAR / scans need SE(3) equivariance for object detection that respects how you mounted the sensor.
Sets and graphs. Permutation equivariance is non-negotiable: GNNs, Deep Sets, transformers (without positional encoding) all sit here.
Medical 3-D volumes. CT / MRI / PET with rotated patient pose. Rotation equivariance helps small-data fine-tunes generalise.

The slogan. Equivariance is data efficiency in disguise. A network that can't rotate has to learn every rotation separately; an equivariant network sees the rotation as a symmetry and spends its capacity on the parts that actually matter.

Step 5

Practical gotchas

Discrete vs continuous group. C4 (90°) and C8 (45°) are easy to implement; continuous SO(2) requires steerable filters or harmonic networks.
Cost. A C4 group conv has 4× the channels of a vanilla conv; train and inference cost scale accordingly. Modern hardware tolerates it.
Combine with augmentation, don't replace it. Equivariant ≠ no augmentation; you still want crops, brightness, noise. The equivariant axis takes one source of variation off the table.
Don't equivary the wrong thing. If your task depends on orientation (text recognition, scene understanding with "up"), rotation equivariance is wrong. Use it where symmetry truly exists.
Libraries. escnn / e2cnn (PyTorch), se3-transformer (PyTorch), e3nn (PyTorch for molecules), jaxon / JAX-MD for SE(3) equivariant work in JAX.