Equivariance

Equivariance means a model output transforms predictably when the input is transformed.

Formally, for a transformation $g$ :

f (g \cdot x) = g \cdot f (x)

The output changes in the same structured way as the input.

More generally, inputs and outputs can transform under different representations:

f (ρ_{in} (g) x) = ρ_{out} (g) f (x)

Here $ρ_{in} (g)$ describes how the input changes and $ρ_{out} (g)$ describes how the output should change.

For 3D coordinates:

x_{i}^{'} = R x_{i} + t, \overset{x}{^}_{i}^{'} = R \overset{x}{^}_{i} + t

where $R \in S O (3)$ and $t \in R^{3}$ . A coordinate predictor should satisfy:

F (R X + 1 t^{⊤}) = R F (X) + 1 t^{⊤}

with the same point ordering.

Why It Matters

Coordinate predictions should rotate or translate with the input structure.
Equivariance reduces the burden of learning symmetries from data alone.
Many geometric neural networks combine equivariant internal features with invariant task outputs.

Translation: shifting all coordinates by the same $t$ .
Rotation: applying $R \in S O (3)$ .
Rigid motion: applying $(R, t) \in S E (3)$ .
Rigid motion with reflection: applying $(R, t)$ with $R \in O (3)$ , often called E(3).
Permutation: reordering atoms, residues, nodes, or set elements.

The model is equivariant to rotations but not translations because absolute coordinates enter hidden features.
The architecture is equivariant but preprocessing uses a ligand-defined frame unavailable at deployment.
Reflection equivariance is enforced even though chirality should be preserved.
A scalar target is treated as equivariant, or a coordinate target is reduced to invariant features too early.

What transformation group is being considered?
Which model outputs should transform, and which should remain invariant?
Is equivariance exact by construction or encouraged by data augmentation?
Does preprocessing introduce a frame that changes the symmetry assumption?
Are vector or coordinate targets transformed consistently during augmentation?
Is the same equivariance contract valid for train, validation, test, and deployment inputs?
Does the coordinate modeling contract state output type, loss, and metric?