Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Groups: Equivariance Beyond Translation

6 minute read

Published:

TL;DR: The previous chapter derived convolution from translation. Nothing in the derivation was specific to translation — it used only that shifts form a group. Replace that group and the machinery still runs: rotations give rotation-equivariant filters, rigid motions of \(\mathbb{R}^3\) give the architectures used for molecules. The cost is precise and computable: weights must be constant on each orbit of the group, so a \(3\times3\) filter under 90° rotations drops from 9 free parameters to exactly 3.

The generalisation

A group \(G\) acting on a domain gives, for each \(g \in G\), a way of transforming a signal. Translation was the case \(G = \mathbb{Z}_n\). A layer \(L\) is equivariant when

\[ L \, \rho(g) \;=\; \rho(g) \, L \qquad \text{for every } g \in G , \]

where \(\rho(g)\) is how \(g\) acts on the representation. For translation this was \(LS = SL\) with a single generator; for a bigger group it is one equation per generator.

The consequence generalises too. Commuting with every group element forces the weights to be constant on orbits — where the orbit of a position is the set of places the group can move it to. Translation on a cycle has one orbit (you can shift anything anywhere), which is why a circulant matrix has one free value per diagonal. Richer groups partition the domain into more orbits, and each orbit contributes one free parameter.

Counting: a 3×3 filter under 90° rotations

Take \(G = C_4\), the four rotations by multiples of 90°, acting on the nine cells of a \(3 \times 3\) filter. Rotation sends the offset \((dx, dy)\) to \((-dy, dx)\). Applying it repeatedly partitions the cells:

orbitcellssize
corners\((-1,-1), (-1,1), (1,-1), (1,1)\)4
edges\((-1,0), (0,-1), (0,1), (1,0)\)4
centre\((0,0)\)1

Three orbits. So a rotation-equivariant \(3 \times 3\) filter has 3 free parameters instead of 9 — one shared by the four corners, one by the four edge cells, one for the centre. The constraint is exact and the arithmetic is checkable: \(4 + 4 + 1 = 9\) cells in \(3\) orbits.

This is the same trade in both directions. Fewer parameters means less to overfit and less to learn — good when the symmetry is real. But those parameters are gone, not merely regularised: a rotation-equivariant filter cannot represent a feature that distinguishes up from down. If your task depends on orientation, you have removed the very capacity you need, and no amount of data recovers it.

Adding reflections gives the eight-element group \(D_4\), which merges orbits further and constrains more. Both appear in the literature as p4 and p4m group convolutions.

Two ways to build them

Group convolution takes the direct route: instead of a feature map over positions, keep one over position–group-element pairs, and convolve over the group. Every rotated copy of a filter is applied, and the responses are carried along as separate channels. Straightforward, and the cost is a factor of \(\lvert G \rvert\) in memory and compute.

Steerable methods avoid materialising the copies. Rather than storing every rotation of a filter, they build filters from a basis on which the group acts in a known way, so a rotation of the input becomes a predictable mixing of the output channels rather than a re-computation. The bookkeeping is heavier — it is where irreducible representations and spherical harmonics enter — but nothing is duplicated.

Continuous groups, and where they matter

For point clouds and molecules the relevant group is continuous: \(SE(3)\), the rotations and translations of 3-D space, or \(E(3)\) if you also allow reflections. Here equivariance is not a convenience, it is a correctness requirement — a predicted force must rotate with the molecule, and a predicted energy must not change at all.

Book II’s geometric subsection covers the architectures: E(n)-equivariant GNNs, SE(3)-Transformers, and tensor field networks. Two points from there that belong here:

  • Energies are invariant, forces are equivariant. Getting these the wrong way round is the standard error, and the blueprint chapter sets out the distinction.
  • \(E(n)\) includes reflections and \(SE(n)\) does not, so an \(E(n)\)-equivariant model is blind to chirality. For molecules, where mirror-image compounds can behave differently, that blindness is a real modelling decision rather than a technicality.
The group must be known in advance, and being wrong about it is expensive. Equivariance is a hard architectural constraint, not a soft prior. Build in a symmetry the data does not have and you have permanently removed the capacity to model the asymmetry — the network cannot learn its way out. Build in a symmetry the data does have and you get a genuine reduction in what must be learned. Conversely, a Transformer assumes almost nothing about which pairs interact, which is why the last chapter finds it winning when data and compute are plentiful.

✅ Key Takeaways

  • Equivariance generalises from \(LS = SL\) to \(L\rho(g) = \rho(g)L\) for every \(g\) in a group. Nothing in the grid derivation was specific to translation.
  • Weights become constant on orbits. Free parameters = number of orbits, and this is exact arithmetic, not an estimate.
  • A \(3\times3\) filter under 90° rotations has 3 orbits — corners, edges, centre — so 3 free parameters instead of 9.
  • Group convolution materialises every transformed filter and costs a factor of \(\lvert G \rvert\); steerable methods avoid the duplication at the price of representation-theoretic bookkeeping.
  • For molecules the group is \(SE(3)\) or \(E(3)\), and equivariance is a correctness requirement: energies invariant, forces equivariant. \(E(n)\) includes reflections, so it cannot see chirality.
  • Equivariance is a hard constraint. Assume the wrong group and the lost capacity cannot be trained back.

References

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