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Causal Discovery via Transformed Low-Rank Quantile Surfaces

NeurIPS 2026 · Probabilistic and scientific modeling › Causal discovery · approach: Constraint-based structure learning with latents · 0 citations (OpenAlex)

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Proposes Low-Rank Quantile Surfaces (LRQS), a bivariate causal model where a monotone-transformed conditional quantile surface is low rank in the causal direction, with identifiability theory and an alternating rank-constrained isotonic causal score.

Problem

Bivariate causal direction inference under distributional asymmetry beyond location-scale noise assumptions, where observed conditional quantile surfaces may not be low rank and the output transformation is unknown.

Approach

Assume that in the causal direction an unknown increasing transformation of the conditional quantile surface has low rank (generalizing LSNM and PNL-HNM), prove generic identifiability via reverse-direction local restrictions on cause log-densities, and score direction by reconstruction error of alternating low-rank plus isotonic fitting.

Key results

• On 42 custom variants, LRQS (est-g) reaches 0.952 average accuracy versus 0.730 for the best baseline (CVEL).
• LRQS (fix-g) runs in about 0.05 s per pair versus about 35 s for QPE-f variants, with 0.740 average accuracy on the 24 benchmarks, tied with QPE-f (poly) at 0.744.
• On the 24 existing benchmarks, LRQS (est-g) averages 0.654 accuracy, below QPE-k (0.741) and QPE-f (poly) (0.744), so the method is not uniformly best on standard benchmarks.

Surprising findings

• On the 24 standard benchmarks, LRQS (est-g) averages 0.654 accuracy, below several baselines, including QPE-k at 0.741.
• Removing the effective rank constraint gives zero reconstruction scores in both directions, so the low-rank constraint is essential for discrimination in the tested settings.

Limitations

• Bivariate only; does not address multivariate graphs, latent confounding, or selection bias.
• Free-basis identifiability theorem is proved only for one reverse component (L=1).
• Practical score depends on conditional quantile estimation, the number of components K_fit, discretization, and a nonconvex alternating fit.

Experimental setup

Datasets: 24 bivariate causal benchmark datasets (AN, LS, SIM, SIM-c, Cha, Net, Per, Sig, Qd-V, NN-V, Tue, D4-s1, among others)

Benchmarks: 24 bivariate causal benchmark datasets custom structural-complexity benchmark (12 variants) custom PNL benchmark (30 variants)

Metrics: causal direction accuracy average execution time (s)

Baselines: ANM DIVOT CVEL QPE-k QPE-f (fixed) QPE-f (poly) QPE-f (lowrank)

Builds on: post-nonlinear heteroscedastic noise models quantile partial effects location-scale noise models additive noise models

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