Invariant Feature

An invariant feature does not change when the input is transformed by a symmetry group.

For a transformation $g\in G$ and feature map $\varphi$:

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

The feature can be used when the task should ignore that transformation.

Examples

• Pairwise distance:

$$d_{ij}=\Vert x_i-x_j{\Vert }_2$$

• Angle between two vectors:

$$\cos \theta =\frac{u^{\top }v}{\Vert u{\Vert }_2\Vert v{\Vert }_2}$$

• Norm of a vector:

$$\Vert v{\Vert }_2$$

• Graph-level readout after permutation-invariant pooling:

$$h_G=\sum _{i=1}^N{h}_i$$

Uses

• Scalar property prediction.
• Binding affinity or energy prediction.
• Classification and ranking.
• Graph-level molecular readout.
• Distance/contact features for 3D structures.

Feature Map

Feature	Invariant To	Not Necessarily Preserved
distance $\Vert x_i-x_j{\Vert }_2$	translation and rotation	direction, chirality, absolute frame
angle $\angle ijk$	translation and rotation	handedness unless signed orientation is included
dihedral magnitude	translation and rotation	sign can matter for stereochemistry
vector norm $\Vert v\Vert$	rotation	vector direction
sorted neighbor distances	permutation within neighbors	identity of each neighbor
graph pooled state	node permutation	local attribution and rare-site signal

Invariance Claim Boundary

Invariant features are appropriate only when the target is also invariant:

$$y(g\cdot x)=y(x)$$

for the transformation group $G$. For scalar molecular properties this is usually true for rotations and translations. For coordinate updates, forces, velocities, and pose refinement, invariant features alone are insufficient unless they are combined with equivariant directions.

Molecular Modeling Risks

Risk	Example
chirality loss	using only pairwise distances can fail to distinguish mirror-image configurations
pose leakage	invariant distances computed from a known bound pose may be unavailable at inference
size leakage	sum features can encode molecule or protein size when size is a dataset artifact
context leakage	pocket features can be ligand-defined rather than deployment-defined
over-invariance	augmentation or feature design removes signal required by the label

Checks

• Which group should the feature be invariant to: translation, rotation, reflection, permutation, or sequence shift?
• Is the target also invariant to that group?
• Does the feature discard information needed for the task, such as chirality or orientation?
• Is the invariant feature computed from public/generic inputs rather than deployment-unavailable context?
• Is the invariant feature used as input, target, metric, or readout?
• Does the feature preserve atom/residue identity and mapping when required?

Related

• Invariance
• Equivariant feature
• Distance geometry
• Pooling and readout
• Stereochemistry
• Coordinate modeling contract
• 3D structure