Topological Residual Asymmetry for Bivariate Causal Direction

Inferring causal direction from purely observational bivariate data is fragile: many methods commit to a direction even in ambiguous or near non-identifiable regimes. We propose Topological Residual Asymmetry (TRA), a geometry-based criterion for additive-noise models. TRA compares the shapes of two cross-fitted regressor-residual clouds after rank-based copula standardization: in the correct direction, residuals are approximately independent, producing a two-dimensional bulk, while in the reverse direction – especially under low noise – the cloud concentrates near a one-dimensional tube. We quantify this bulk-tube contrast using a 0D persistent-homology functional, computed efficiently from Euclidean MST edge-length profiles. We prove consistency in a triangular-array small-noise regime, extend the method to fixed noise via a binned variant (TRA-s), and introduce TRA-C, a confounding-aware abstention rule calibrated by a Gaussian-copula plug-in bootstrap. Extensive experiments across many challenging synthetic and real-data scenarios demonstrate the method’s superiority.

💡 Research Summary

This paper tackles the notoriously fragile problem of determining causal direction from purely observational bivariate data. Existing approaches—such as RESIT, IGCI, RECI, SLOPPY, KCDC, and various supervised or generative methods—often commit to a direction even when the underlying model is near‑non‑identifiable (e.g., close to linear‑Gaussian) or when sample size is limited. The authors propose a novel geometry‑based criterion called Topological Residual Asymmetry (TRA) that exploits the shape of residual point clouds generated by additive‑noise models (ANMs).

In an ANM, the true causal direction satisfies Y = f(X) + ε with ε ⟂ X. After fitting regressors in both directions and computing out‑of‑sample residuals, the residual cloud in the correct direction should be (asymptotically) independent of the predictor, yielding a two‑dimensional “bulk” distribution after a rank‑based copula transform. In the opposite direction, unless the model falls into a special non‑identifiable case, the residuals remain dependent on the predictor; with low noise they become nearly deterministic, concentrating around a one‑dimensional curve (“tube”). TRA quantifies the bulk‑tube contrast by applying 0‑dimensional persistent homology to the copula‑standardized clouds. Concretely, the authors compute the Euclidean minimum spanning tree (MST) of each cloud, extract the edge‑length multiset, and evaluate a functional TP