Causal Discovery via Transformed Low-Rank Quantile Surfaces
Low-Rank Quantile Surfaces model causal direction via monotone transformations yielding low-rank quantile surfaces, proving generic identifiability and outperforming location-scale methods on nonlinear, heteroscedastic data.
Published 2026Paris Poster Session 2 · Wed, Dec 9, 5:00 PM–7:00 PM local time · Paris Poster HallarXiv ↗OpenReview ↗
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LRQS earns its lecture with elegant isotonic low-rank quantile surfaces that break location-scale limits, though it remains an unreleased synthetic benchmark without real-world partners, code, or scaling proof.
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.