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Topological Residual Asymmetry for Bivariate Causal Direction

Mouad El Bouchattaoui

TL;DR

The paper tackles the challenge of inferring causal direction from purely observational bivariate data under additive-noise models. It introduces Topological Residual Asymmetry (TRA), a geometry-based criterion that compares copula-standardized forward and reverse residual clouds using a 0D persistent-homology proxy derived from the MST edge lengths, capturing a bulk vs tube distinction between the two directions. TRA is extended with TRA-s for fixed-noise regimes and TRA-C, a confounding-aware abstention rule calibrated by a Gaussian-copula bootstrap, providing controlled decisions in the presence of latent confounding. The authors establish consistency in a small-noise triangular-array setting, justify mesoscopic smoothing to recover separation under fixed noise, and demonstrate robust performance across synthetic stress tests and real-world Tübingen pairs, while emphasizing abstention to avoid unwarranted causal claims. Overall, TRA offers a principled, topology-guided approach to causal orientation with calibrated uncertainty and empirical superiority in diverse scenarios.

Abstract

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.

Topological Residual Asymmetry for Bivariate Causal Direction

TL;DR

The paper tackles the challenge of inferring causal direction from purely observational bivariate data under additive-noise models. It introduces Topological Residual Asymmetry (TRA), a geometry-based criterion that compares copula-standardized forward and reverse residual clouds using a 0D persistent-homology proxy derived from the MST edge lengths, capturing a bulk vs tube distinction between the two directions. TRA is extended with TRA-s for fixed-noise regimes and TRA-C, a confounding-aware abstention rule calibrated by a Gaussian-copula bootstrap, providing controlled decisions in the presence of latent confounding. The authors establish consistency in a small-noise triangular-array setting, justify mesoscopic smoothing to recover separation under fixed noise, and demonstrate robust performance across synthetic stress tests and real-world Tübingen pairs, while emphasizing abstention to avoid unwarranted causal claims. Overall, TRA offers a principled, topology-guided approach to causal orientation with calibrated uncertainty and empirical superiority in diverse scenarios.

Abstract

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.
Paper Structure (94 sections, 12 theorems, 165 equations, 5 figures, 1 algorithm)

This paper contains 94 sections, 12 theorems, 165 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Theoretical Setup
  4. Copula standardization.
  5. Mesoscopic scale: separating "bulk" from "curve".
  6. Core intuition.
  7. Remarks.
  8. Fixed Noise Regime
  9. Assumptions for TRA-s under fixed noise.
  10. TRA-C: abstention under unobserved confounding via a Gaussian-copula plug-in bootstrap
  11. Gaussian-copula null and Gaussianization.
  12. TRA-C calibration.
  13. Experiments
  14. Baselines and decision rule.
  15. Metrics.
  16. ...and 79 more sections

Key Result

Theorem 3.5

Assume the small-noise ANM $Y_n=f(X)+\sigma_n\varepsilon$ with $\sigma_n\downarrow 0$ and Assumptions ass:A_model--ass:A_scale_est. Then Hence $\Delta_n\xrightarrow{\mathbb P}1$. In particular, for any deterministic threshold $\tau_n\downarrow 0$, the abstaining rule $\widehat{\mathrm{dir}}_n=X\to Y$ if $\Delta_n>\tau_n$, $Y\to X$ if $\Delta_n<-\tau_n$, and abstain otherwise, satisfies

Figures (5)

  • Figure 1: Bulk-tube contrast in copula transform. Forward clouds are 2D, while reverse clouds collapse near a curve. $Y_{n,i} \mid X_i \sim \mathcal{N}(X_i^3,n^{-1/2}), n=250$
  • Figure 2: TRA-C calibration on a Gaussian-copula confounding null. Rank-Gaussianized data fit a correlation $\widehat{\rho}$ and induce a Gaussian-copula bootstrap for the score $\Delta_n$.
  • Figure 3: Evolution of coverage under (non)linear latent confounding as function of stress parameters (top) and sample size (bottom). Lower is better.
  • Figure 4: Synthetic ANM atlas. Directed risk vs. sample size $n$ and the scenario stress parameter. TRA-s is consistently low-risk across regimes; baselines show regime-specific failures.
  • Figure 5: Results on Tübingen cause--effect pairs. Coverage, accuracy conditional on deciding, and directed risk for all methods.

Theorems & Definitions (35)

  • Definition 3.1: TRA score and direction
  • Theorem 3.5: Small-noise TRA consistency (normalized score)
  • Theorem 3.9: Fixed-noise TRA-s consistency
  • Theorem 3.10: TRA-C: asymptotic level under a plug-in Gaussian-copula bootstrap
  • Definition 1.1: Law / pushforward
  • Definition 1.2: Coupling
  • Definition 1.3: Big-$O$ and little-$o$ in probability
  • Remark 1.4: Basic algebra
  • Definition 1.5: Bounded--Lipschitz metric
  • Remark 1.6: Coupling bound and finiteness
  • ...and 25 more