Latent Confounded Causal Discovery via Lie Bracket Geometry

2026-06-17

AI Summary

This work addresses the challenge of accurately reconstructing true causal structures in the presence of latent confounding, a setting where existing causal discovery methods often fail. The authors propose a novel paradigm grounded in the geometry of intervention manifolds, uniquely integrating categorical models of interventions with Lie bracket geometry. By leveraging Frobenius residuals to directly detect non-integrability induced by latent variables, and combining Radon–Nikodym derivative estimation with spectral decomposition to construct geometric residuals, the approach yields two new algorithms—BRIDGE and SKFM. These algorithms dramatically reduce the search space for directed acyclic graphs (DAGs) by multiple orders of magnitude while achieving high recall in latent-confounded scenarios, thereby opening a new pathway toward differentiable causal inference.

Abstract

Recent work on Kan-Do-Calculus (KDC) has established that the boundary between passive observation and active intervention in causal inference is a category-theoretic bi-adjunction, with interventions modeled by left Kan extensions and conditioning by right Kan extensions. This paper introduces two causal discovery algorithms under latent confounding, building on the information-geometric and categorical consequences of KDC. In smooth statistical settings, Radon-Nikodym derivatives between observational and interventional measures induce local causal vector fields; failures of these fields to close under Lie brackets become computable Frobenius residuals, which we interpret as witnesses of failed visible integrability and possible latent or unmodeled structure. Our first algorithm, BRIDGE (Bracket Residuals for Interventional Discovery and Geometric Estimation), combines an interventional density or Radon-Nikodym-ratio engine with a geometric screen that proposes a high-recall family of admissible arrows, identifies non-closing visible pairs as latent-obstruction candidates, and passes the reduced family to downstream score-based or differentiable discovery routines. The second algorithmic contribution, Spectral Kan-Do Flow Matching (SKFM), learns amortized intervention fields and factors latent curvature spectrally, exposing the direct Lie-space endpoint toward which BRIDGE points. A detailed set of experiments show that both algorithms are capable of discovering causal models with latent confounders while collapsing the super-exponential space of possible DAGs by many orders of magnitude. This paper introduces a new paradigm in causal discovery, where latent structure is inferred directly from the geometry of intervention-induced flows.

Problem

• causal discovery
• latent confounding
• intervention
• Lie bracket
• causal inference

Innovation

• Lie bracket geometry
• latent confounding
• Kan extension
• causal discovery
• interventional flows