Equivariance Testing

Equivariance Testing is a model validation technique that verifies whether the model's output transforms predictably when the input is transformed — unlike invariance (output unchanged), equivariance means the output changes in a corresponding, predictable way (e.g., rotating input rotates the output mask).

Invariance vs. Equivariance

• Invariance: $f(T(x)) = f(x)$ — output is unchanged by the transformation.
• Equivariance: $f(T(x)) = T'(f(x))$ — output transforms correspondingly with the input transformation.
• Example: Classification should be rotation-invariant. Segmentation should be rotation-equivariant.
• Testing: Apply transformation $T$ and verify the output-transform relationship holds.

Why It Matters

• Segmentation/Detection: Object detection and segmentation models should be equivariant to geometric transforms.
• Physics: Physical models should be equivariant to coordinate transformations (rotation, translation).
• Architecture Design: Equivariance testing validates that architectures (group-equivariant CNNs, E(n)-equivariant networks) achieve the desired symmetries.

Equivariance Testing is testing that outputs transform correctly — verifying that model outputs respond predictably to input transformations.