Equivariant Feature

An equivariant feature transforms predictably when the input is transformed.

For a transformation $g\in G$, input action $g\cdot x$, and output representation $\rho (g)$:

$$\varphi (g\cdot x)=\rho (g)\varphi (x)$$

For 3D rotations, a vector feature $v_i\in {\mathbb{R}}^3$ usually transforms as:

$$v_i^{\prime }=R{v}_i$$

where $R\in SO(3)$.

Examples

• Coordinate:

$$x_i^{\prime }=R{x}_i+t$$

• Relative direction:

$$u_{ij}=\frac{x_j-x_i}{\Vert x_j-x_i{\Vert }_2},u_{ij}^{\prime }=R{u}_{ij}$$

• Coordinate update:

$$x_i^{(t+1)}=x_i^{(t)}+\Delta x_i,\Delta x_i^{\prime }=R\Delta x_i$$

• Vector field, force, velocity, or score direction.

Why It Matters

Equivariant features let a model predict geometry without learning arbitrary coordinate-frame rules from data. This is useful for structure prediction, docking pose generation, conformer generation, molecular dynamics surrogates, and 3D generative models.

Feature Map

Feature	Transform Rule	Use
coordinate $x_i$	$x_i^{\prime }=R{x}_i+t$	point position, pose, conformer, structure
relative vector $r_{ij}=x_j-x_i$	$r_{ij}^{\prime }=R{r}_{ij}$	direction-aware message passing
unit direction $u_{ij}$	$u_{ij}^{\prime }=R{u}_{ij}$	normalized geometric direction
force $F_i$	$F_i^{\prime }=R{F}_i$	dynamics, energy gradients
velocity $v_i$	$v_i^{\prime }=R{v}_i$	flow matching, coordinate ODE
tensor feature	$T^{\prime }=\rho (R)T$	higher-order angular information

Scalar-Vector Separation

A common pattern is to compute invariant scalar weights and apply them to equivariant directions:

$$\Delta x_i=\sum _{j\in \mathcal{N}(i)}{a}_{ij}(x_j-x_i),a_{ij}=\varphi (h_i,h_j,\Vert x_i-x_j{\Vert }_2,e_{ij})$$

If $a_{ij}$ is invariant and the direction rotates, then $\Delta x_i$ rotates. This gives a simple route to equivariant coordinate updates without putting absolute coordinates into scalar features.

Readout Boundary

Target	Correct Output Type
affinity, energy, probability, class, rank	invariant scalar
force, velocity, score field, displacement	equivariant vector
coordinates, pose, conformer, backbone	equivariant coordinates plus translation behavior
distance, contact, RMSD-like diagnostic	invariant scalar derived from coordinates
orientation-sensitive chirality	preserve orientation information; reflection behavior must be explicit

Checks

• What representation does each feature live in: scalar, vector, tensor, or higher-order irrep?
• Does the output need translation equivariance, rotation equivariance, or both?
• Are reflections valid for this domain, or does chirality require orientation-preserving transforms?
• Are equivariant features converted to invariant readouts before scalar prediction?
• Is every preprocessing step equivariant, or only the neural layer?
• Are coordinate augmentations applied consistently to input, target, vector labels, and masks?
• Are units and coordinate frames fixed before comparing models?

Related

• Equivariance
• Invariant feature
• Coordinate frame
• Coordinate update
• Coordinate modeling contract
• Irreducible representation
• Equivariant GNN