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Causal Abstraction Learning based on the Semantic Embedding Principle
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Proposes a category-theoretic, data-driven framework for learning linear causal abstractions under the semantic embedding principle, casting it as Riemannian optimization on the Stiefel manifold with three algorithms (LinSEPAL-ADMM, LinSEPAL-PG, CLinSEPAL), evaluated on synthetic and brain rs-fMRI data.
Problem
Learning causal abstraction (CA) maps between high- and low-level SCMs when SCMs are inaccessible, interventional data is unavailable, and samples are misaligned, using only partial prior knowledge of the abstraction structure.
Approach
Introduces the semantic embedding principle (SEP): the high-level distribution must lie on a subspace of the low-level one, so the linear abstraction matrix is constrained to the Stiefel manifold; CA learning becomes minimizing a Gaussian KL divergence over Stiefel-constrained (and, for CLinSEPAL, support-constrained) matrices.
Key results
- With full prior knowledge, all three methods return constructive CAs for all 30 synthetic abstractions per setting and reach good KL alignment; CLinSEPAL and LinSEPAL-ADMM are slightly better than LinSEPAL-PG.
- With partial prior knowledge (4,2 setting), only CLinSEPAL consistently returns constructive linear CAs; LinSEPAL-PG fails the 90% constructiveness threshold at 50% prior drop.
- On brain data with full prior knowledge, CLinSEPAL recovers the ground-truth abstraction V*; with uncertain 8-network prior, it gives low misalignment at low uncertainty but makes structural mistakes at medium/high uncertainty.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CLinSEPAL rather than LinSEPAL-PG or LinSEPAL-ADMM with partial structural prior | Linear CA learning · Synthetic data, partial prior knowledge | constructiveness | Helped |
| Reduce the amount of structural prior knowledge | CLinSEPAL · Synthetic data, partial prior knowledge | constructiveness | Mixed |
| Reduce structural prior knowledge to 50% | LinSEPAL-PG · Synthetic data, partial prior knowledge | constructiveness | Hurt |
| Reduce structural prior knowledge to 50% | LinSEPAL-ADMM · Synthetic data, partial prior knowledge | constructiveness | Hurt |
| Use CLinSEPAL or LinSEPAL-ADMM instead of LinSEPAL-PG | Linear CA learning · Synthetic data, full prior knowledge | normalized absolute Frobenius distance | Mixed |
| Increase the high-level dimension across the tested coarse-graining settings | Proposed methods · Synthetic data, full prior knowledge | normalized absolute Frobenius distance | Helped |
| Use CLinSEPAL with low uncertainty in the structural prior | Brain network CA learning · Brain data, partial prior knowledge, 8 functional networks | CA structure recovery | Helped |
| Increase uncertainty in the structural prior | CLinSEPAL · Brain data, partial prior knowledge, 8 functional networks | CA structure recovery | Hurt |
Surprising findings
- At high coarse-graining (h=2), the learned V can be far from ground truth in Frobenius distance even though its support (F1) is correct, because the KL kernel is large.
Limitations
- Convergence guarantee for Riemannian ADMM is not proven, so LinSEPAL-ADMM lacks a theoretical guarantee.
- Only linear abstractions with Gaussian measures and KL divergence are solved; nonlinear and other measures are left as future work.
- Partial prior knowledge can reduce constructiveness and accuracy under high uncertainty, and the KL objective has non-unique solutions.
Experimental setup
Datasets: Human Connectome Project rs-fMRI (100 adults, 89 ROIs) synthetic Gaussian covariance data
Benchmarks: synthetic linear CA settings (l,h) in {(12,2),(12,4),(12,6),(4,2)} brain rs-fMRI causal abstraction (45 to 14 and 45 to 8 ROIs)
Metrics: constructiveness fraction KL divergence normalized Frobenius distance F1 score of support
Baselines: LinSEPAL-ADMM LinSEPAL-PG
Builds on: ADMM manifold proximal gradient splitting of orthogonality constraints alpha-abstraction framework
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