Quantile Geometry: Finite-Rank Drift Ratios for Causal Discovery

Unknown nonlinear transformations of an effect can obscure the structure used to distinguish cause from effect. We study the drift ratio, which measures how the effect’s conditional quantiles move as the cause changes, relative to their local spacing. This quantity is unchanged by smooth increasing transformations of the effect. We introduce finite-rank drift-ratio (FRDR) models, which assume that the drift-ratio surface has finite rank, meaning that its variation is captured by finitely many shared functions of the quantile level. The framework includes additive noise models, post-nonlinear models, location-scale noise models, and post-nonlinear heteroscedastic noise models. Under suitable conditions, we prove generic identifiability of the causal direction in FRDR models: the reverse drift ratio has rank greater than any fixed finite bound, apart from exceptional distributions of the cause. We derive a quantile geometry (QG) score, QG-FRDR, that applies across these model classes, along with four specialized scores that exploit their additional structure. These scores require neither noise reconstruction nor estimation of the unknown transformation. Experiments demonstrate fast computation for QG-FRDR and strong directional accuracy for its weak formulation on structural-model simulations. The class-specific QG-ANM, QG-LSNM, and QG-PNL scores achieve high accuracy within their target classes.