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Identification of Latent Confounders via Investigating the Tensor Ranks of the Nonlinear Observations

ICML 2025 · Probabilistic and scientific modeling › Causal discovery · approach: Latent-variable causal estimation

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Extends the tensor rank condition to discrete latent structure models with mixed continuous/discrete observed variables by discretizing data until full-rank holds, and proposes a two-stage algorithm (causal clusters, then latent structure via CI tests) that identifies latent causal structure up to Markov equivalence.

Problem

Learning discrete latent variable causal structures from mixed-type (continuous and discrete) observational data, where the tensor rank condition was previously only applicable to fully discrete observations.

Approach

Under the completeness condition, a discretization of continuous observed variables can be found that satisfies the full-rank assumption, so the tensor rank condition (with d-separation implications) applies to mixed-type data; a rank-stop-increasing discretization procedure implements this, and a two-sufficient-measurement condition supports a two-stage identification algorithm allowing multiple

Key results

• On simulated mixed-type data across Case I-III, the proposed method consistently gives the best latent omission and F1 scores versus TS-PC and Mixed-LT in most scenarios.
• TS-PC performs poorly because random discretization can violate the full-rank assumption.
• Theorem 4.19 shows the latent causal structure is identified up to a Markov equivalence class under full-rank and completeness conditions.

Measured effects reported in the paper

Surprising findings

• Randomly discretizing continuous variables can break the full-rank assumption and cause the tensor rank method to misidentify latent structure.

Limitations

• Assumes purity (no direct edges between observed variables) and three-pure-children structure, with extension to impure structures left to future work.
• Requires completeness condition and full-rank assumption, which cannot be directly tested since P(X|L) is latent.
• Simulations use only small sample sizes (5k to 50k in text) and a limited set of structures.

Experimental setup

Datasets: simulated mixed-type data from discrete contingency tables and Gaussian mixture models real-world data (unspecified in text excerpt)

Benchmarks: simulated structures Case I-III

Metrics: latent omission F1 score

Baselines: TS-PC (PC-TENSOR-RANK) Mixed-LT (mixed latent tree, Zhou et al., 2020)

Builds on: tensor rank condition (Chen et al., 2024) PC-TENSOR-RANK pure measurement model / latent structure learning (Silva et al., 2006) mixture-model identifiability (Kivva et al., 2021)

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