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Linear Causal Representation Learning from Unknown Multi-node Interventions
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Establishes identifiability for linear causal representation learning under unknown multi-node stochastic interventions, giving perfect identifiability with hard interventions and identifiability up to ancestors with soft interventions, via the score-based UMNI-CRL algorithm.
Problem
Interventional causal representation learning has mostly assumed single-node interventions, but in practice the set of intervened nodes per environment is unknown and can contain multiple nodes; identifiability and algorithms for this unknown multi-node setting are open.
Approach
Uses score differences between interventional and observational environments and linearly aggregates them (with integer mixing vectors) to synthesize score differences for hypothetical intervention sets, recovering the linear encoder in stages (causal order, ancestors, then unmixing for hard interventions) under a regularity condition on multi-node interventions.
Key results
- Under UMN soft interventions with n=4, d=10, SHD to transitive closure is 0.91 and MCC 0.95; MCC stays above 0.90 for all settings up to n=8.
- Under UMN hard interventions, MCC is 0.98 at n=4 and 0.87 at n=8 (d=10), with SHD to the true DAG growing from 0.75 to 9.70.
- Theoretical results show perfect identifiability with hard UMN interventions using n+1 environments, and identifiability up to ancestors with soft UMN interventions for general latent models and linear transformations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase observed dimension from d=10 to d=50 | UMNI-CRL · UMN soft and hard interventions | SHD: $0.91 \pm 0.12$ -> $0.77 \pm 0.12$ (soft, n=4); $0.75 \pm 0.11$ -> $0.66 \pm 0.10$ (hard, n=4) | No clear effect |
| Increase latent graph size from n=4 to n=8 | UMNI-CRL · UMN soft interventions, d=10 | SHD: $0.91 \pm 0.12$ -> $7.63 \pm 0.41$ | Hurt |
| Increase latent graph size from n=4 to n=8 | UMNI-CRL · UMN hard interventions, d=10 | SHD: $0.75 \pm 0.11$ -> $9.70 \pm 0.52$ | Hurt |
| Increase latent graph size from n=4 to n=8 | UMNI-CRL · UMN soft interventions, d=10 | MCC: $0.95 \pm 0.01$ -> $0.89 \pm 0.01$ | Hurt |
| Increase latent graph size from n=4 to n=8 | UMNI-CRL · UMN hard interventions, d=10 | MCC: $0.98 \pm 0.02$ -> $0.87 \pm 0.03$ | Hurt |
| Use UMNI-CRL for latent variable recovery under UMN interventions | UMNI-CRL · Linear Gaussian SEM, UMN soft and hard interventions | MCC | Helped |
Surprising findings
- Identifiability with strongly separating sets of UMN stochastic hard interventions is impossible, despite such sets sufficing for do-interventions.
Limitations
- Assumes linear transformation from latent to observed space.
- Search-based aggregation has worst-case exponential complexity in the number of latent nodes.
- Necessary conditions for identifiability under UMN interventions remain open.
Experimental setup
Datasets: synthetic linear Gaussian SEMs with Erdos-Renyi graphs
Benchmarks: synthetic linear Gaussian SEM simulations
Metrics: structural Hamming distance mean correlation coefficient incorrect mixing entry ratio
Baselines: Squires et al. 2023 linear CRL strongly separated multi-node do-intervention CRL linear non-Gaussian multi-node soft intervention CRL
Builds on: score-based CRL for single-node interventions linear CRL with single-node interventions
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