Causal Discovery via Transformed Low-Rank Quantile Surfaces
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 nonparametric causal score using a 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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BibTeX
@misc{kamimura2026causal,
title = {{Causal Discovery via Transformed Low-Rank Quantile Surfaces}},
author = {Ryo Kamimura and Thong Pham},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16931},
eprint = {2609.16931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16931},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Causal Discovery via Transformed Low-Rank Quantile Surfaces
AU - Ryo Kamimura
AU - Thong Pham
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.16931
UR - https://arxiv.org/abs/2609.16931
AB - 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 nonparametric causal score using a 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.
ER -