Causal Discovery via Transformed Low-Rank Quantile Surfaces

arXiv:2609.16931 · stat.ME, cs.AI, cs.LG, stat.ML · Submitted 2026-09-15 · Read on arXiv

stat.ME, cs.AI, cs.LG, stat.ML

Submitted: 2026-09-15

Updated: 2026-09-15

Comments: 25 pages

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

The gist: We propose Low-Rank Quantile Surfaces (LRQS), a bivariate causal model in which, in the causal direction, an unknown monotone transformation of the conditional quantile surface admits a low-rank

Terminology

Abstract

We propose Low-Rank Quantile Surfaces (LRQS), a bivariate causal model in which, in the causal direction, an unknown monotone transformation of the conditional quantile surface admits a low-rank functional decomposition. LRQS subsumes location-scale noise models and post-nonlinear heteroscedastic noise models, while allowing multiple quantile bases to represent changes beyond location-scale effects. We prove generic identifiability of LRQS: the transformed quantile surface is low rank in the causal direction, whereas reverse representability under the corresponding constraints occurs only for exceptional, fine-tuned cause marginals. We provide a simple-yet-powerful causal score using a nonparametric fitting procedure that alternates between rank-constrained approximation of discretized quantile surfaces and isotonic estimation of the unknown monotone transformation. Experiments on synthetic mechanisms with higher-rank distributional shape variation and strong nonlinear distortions, together with standard bivariate benchmarks, show that LRQS is especially effective when conditional distributional shape or observation distortion goes beyond existing location-scale assumptions.

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