Research map / Probabilistic and scientific modeling / Causal discovery
Learning Discrete Latent Variable Structures with Tensor Rank Conditions
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Establishes a tensor-rank graphical criterion for discrete latent structure models and uses it in an iterative algorithm (causal clusters plus PC-style CI tests) to identify measurement and structure models of discrete latent variables.
Problem
Identifying the causal structure among discrete latent variables when relations are nonlinear and the latent structure is not restricted to trees or pyramids.
Approach
Links the tensor rank of the joint contingency table of observed variables to the minimal support of a d-separating conditional set; uses this rank to find causal clusters (measurement model) and to run CI tests among latent variables (structure model) without structural restrictions on latents.
Key results
- On simulated SM1+MM1 at 50k samples, the proposed method reaches 0.00 latent omission, 0.00 latent commission and 0.00 mismeasurement, matching the best baselines.
- On Collider+MM1 at 50k samples, the proposed method attains 0.00 edge omission and 0.00 edge commission, while BayPy and BPC show 1.00 edge omission.
- On SM3+MM1 at 5k samples, the proposed method has 0.25 latent omission versus 0.30 for BayPy and 0.55 for LTM, though its performance is not uniformly best in all settings.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use tensor-rank-based cluster discovery instead of BayPy | learning pure measurement models · SM3 + MM1, 5k | latent omission: 0.30(6) -> 0.25(6) | Helped |
| Use tensor-rank-based cluster discovery instead of BayPy | learning pure measurement models · SM2 + MM2, 5k | latent omission: 0.19(6) -> 0.12(2) | Helped |
| Use tensor-rank-based cluster discovery instead of BayPy | learning pure measurement models · SM1 + MM1, 5k | mismeasurements: 0.00(0) -> 0.05(1) | Hurt |
| Use tensor-rank-based structure learning instead of LTM | learning latent causal structures · Collider + MM1, 5k | edge omission: 0.26(8) -> 0.00(0) | Helped |
| Use tensor-rank-based structure learning instead of BayPy | learning latent causal structures · Collider + MM1, 5k | orientation omission: 1.00(10) -> 0.10(1) | Helped |
| Use tensor-rank-based structure learning instead of BayPy | learning latent causal structures · Star + MM1, 5k | edge commission: 0.00(0) -> 0.20(5) | Hurt |
| Increase sample size from 5k to 50k | Our measurement-model algorithm · SM3 + MM1 | latent omission: 0.25(6) -> 0.08(3) | Helped |
| Increase sample size from 5k to 50k | Our structure-learning algorithm · SM3 + MM1 | edge omission: 0.22(7) -> 0.05(2) | Helped |
| Increase sample size from 5k to 50k | Our structure-learning algorithm · Star + MM1 | edge commission: 0.20(5) -> 0.05(2) | Helped |
Surprising findings
- Tensor rank of a two-way contingency table of any two observed variables gives the latent support size directly, without needing the full structure.
Limitations
- Assumes pure measurement model (no edges among observed variables) and at least three pure children per latent variable.
- Requires all latent variables to share the same number of categories in the main analysis; the sufficient observation assumption restricts latent support size relative to observed support.
- Impure structures and hierarchical latent structures are left to future work.
Experimental setup
Datasets: political efficacy dataset (1719 cases, USA sample) depress dataset (204 samples, twelve observed variables)
Benchmarks: simulated discrete LSM structures (SM1-SM3, Collider, Star) with MM1/MM2 measurement models political efficacy dataset depress dataset
Metrics: latent omission latent commission mismeasurement edge omission edge commission orientation omission
Baselines: BPC LTM BayPy
Builds on: PC algorithm FCI algorithm Building Pure Clusters Bayesian pyramid model CP decomposition
Closest papers by meaning
- Identification of Latent Confounders via Investigating the Tensor Ranks of the Nonlinear Observations (ICML 2025 · similarity 0.96)
- A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables (ICLR 2024 · similarity 0.91)
- Differentiable Causal Discovery for Latent Hierarchical Causal Models (ICLR 2025 · similarity 0.90)
- Identification of Causal Structure with Latent Variables Based on Higher Order Cumulants (AAAI 2024 · similarity 0.90)
- Local Causal Structure Learning in the Presence of Latent Variables (ICML 2024 · similarity 0.89)
- Differentiable Structure Learning and Causal Discovery for General Binary Data (NeurIPS 2025 · similarity 0.89)
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