The unlearning isomorphism is the central theoretical contribution of Towards Certified Shortcut Unlearning in Medical Imaging. It establishes a formal set-theoretic equivalence between correcting a coarse segmentation mask and forgetting specific training data points in the machine unlearning framework.
The key insight relies on the monotonicity of the dilation operator. Consider transitioning from a coarse mask Y^(r1) to a finer mask Y^(r2) where r1 < r2. Due to dilation monotonicity, the set of annotated pixels in the finer mask is strictly a subset of those in the coarse mask. This nesting property allows identifying the “correction” of a mask with the “forgetting” of dilation artefacts — pixels labelled as foreground by the coarse mask but background by the fine mask.
Formally, the atomic data space is defined as Z = X x Lambda, where Lambda is the pixel grid. The forget set is:
D_f := D \ D^(r2) = {z_{n,i,j} in D | (Y_n^(r2))_{ij} = 0}
This is exactly the set of pixel-level data points whose labels “flip” from foreground to background when moving to the finer mask. Mechanically, unlearning these data points corresponds to removing the spurious foreground labels.
Key Details
- This isomorphism ensures that any (epsilon, delta)-certified unlearning guarantee (e.g., from Koloskova et al. (2025)) directly applies to the segmentation refinement setting
- No additional assumptions beyond the standard unlearning framework are needed
- Enables leveraging the full machinery of certified unlearning theory (gradient-based certificates, noise injection) for pixel-level mask correction
- Relies on the single-class assumption: each image contains at most one non-background pathology class