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Pooling Image Datasets with Multiple Covariate Shift and Imbalance
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Proposes a category-theory-inspired framework that learns functor-based latent spaces with linear morphisms to handle many covariate shifts and imbalances when pooling image datasets, replacing multi-stage pipelines.
Problem
Pooling small multi-site imaging datasets with shifts and imbalances in many covariates (age, sex, APOE, scanner) breaks standard invariant representation learning, which handles at most two covariates and needs multi-stage training.
Approach
Model covariate changes as morphisms in a category, represent each covariate change as a learned linear map (orthogonal) in latent space, and train one end-to-end model with reconstruction, prediction and structure-preserving losses, yielding both invariance (same-covariate pairs) and equivariance (different-covariate pairs) without per-pair stages.
Key results
- On ADNI, the invariant model reduces MMD to 0.02 (x10^2 scale) versus 0.16-0.27 for baselines while accuracy is 81%, close to naive 80%.
- On ADCP, the invariant model reaches 86% accuracy and MMD 0.34 compared with 0.70-0.88 for most baselines.
- Single-covariate and 5-covariate equivariance keep accuracy around 80-82% on ADNI with MMD around 0.11.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add scanner invariance to the pooled classifier | Ours · ADNI | MMD (× 10²): 27 -> 02 | Helped |
| Add scanner invariance to the pooled classifier | Ours · ADCP | accuracy: 83 -> 86 | Helped |
| Add scanner invariance to the pooled classifier | Ours · German | MMD (× 10²): 7.7 -> 0.7 | Helped |
| Add gender invariance to the pooled classifier | Ours · Adult | MMD (× 10²): 9.8 -> 0.6 | Helped |
| Add equivariance to one covariate alongside scanner invariance (ablation) | Ours · ADNI | accuracy: 80 -> 82 | Helped |
| Add equivariance to two covariates instead of one (ablation) | Ours · ADNI | MMD (× 10²): 11 -> 10 | Helped |
| Add equivariance to five covariates alongside scanner invariance (ablation) | Ours · ADNI | accuracy: 82 -> 80 | Hurt |
| Increase the invariance loss multiplier from $0.001$ to $0.1$ at latent size $16$ (ablation) | Ours · ADNI | MMD: $18{\scriptstyle(4.4)}$ -> $02{\scriptstyle(2.4)}$ | Helped |
| Increase the invariance loss multiplier from $0.001$ to $0.1$ at latent size $16$ (ablation) | Ours · ADNI | accuracy: $81{\scriptstyle(2.3)}$ -> $81{\scriptstyle(4.0)}$ | No clear effect |
| Increase the latent dimension from $16$ to $64$ with invariance multiplier $0.001$ (ablation) | Ours · ADNI | accuracy: $81{\scriptstyle(2.3)}$ -> $84{\scriptstyle(3.9)}$ | Helped |
| Remove the orthogonality constraint on the latent morphism (ablation) | MNIST addition model · MNIST | inverse-transform image quality | Hurt |
Surprising findings
- Invariance and equivariance constraints did not reduce accuracy and sometimes improved generalization.
- Multi-covariate equivariance (2-covariate) did not produce higher MMD than single covariate, contrary to expectation that it is harder.
Limitations
- Uses simple modules (linear morphisms, ResNet) chosen to demonstrate the formalism rather than maximize accuracy.
- Incompatible with skip-connection architectures such as U-Net because shortcuts bypass the latent constraints.
- Evaluated on small datasets (449 ADNI scans) with 5-fold cross-validation.
Experimental setup
Datasets: ADNI ADCP German credit Adult MNIST
Benchmarks: ADNI ADCP German Adult MNIST
Metrics: accuracy MMD adversarial scanner accuracy (ADV) minimum distance cosine similarity
Baselines: Naive MMD CAI SS RM GE
Builds on: CycleGAN SimCLR group equivariance (GE) MMD invariance CAI
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