HOLOGRAPH: Active Causal Discovery via Sheaf-Theoretic Alignment of Large Language Model Priors
Hyunjun Kim
TL;DR
HOLOGRAPH reframes causal discovery as a presheaf coherence problem, modeling local LLM-derived beliefs on variable subsets as sections of a presheaf $ $ over the power set and treating global coherence as the existence of a global section satisfying descent. It introduces Algebraic Latent Projection to handle latent confounders, a natural gradient optimization scheme with Tikhonov stabilization, and Expected Free Energy–based active querying to efficiently leverage LLM priors. The framework is grounded by four presheaf axioms, with empirical results showing Identity, Transitivity, and Gluing holding to $<10^{-6}$ error across graph sizes, while Locality reveals non-local latent coupling and challenges the classical sheaf assumption. Practically, HOLOGRAPH achieves competitive causal discovery performance on datasets with 50–100 variables and offers a rigorous mathematical lens for integrating LLM priors, although locality violations motivate exploring non-commutative cohomology and hybrid intervention strategies in future work.
Abstract
Causal discovery from observational data remains fundamentally limited by identifiability constraints. Recent work has explored leveraging Large Language Models (LLMs) as sources of prior causal knowledge, but existing approaches rely on heuristic integration that lacks theoretical grounding. We introduce HOLOGRAPH, a framework that formalizes LLM-guided causal discovery through sheaf theory--representing local causal beliefs as sections of a presheaf over variable subsets. Our key insight is that coherent global causal structure corresponds to the existence of a global section, while topological obstructions manifest as non-vanishing sheaf cohomology. We propose the Algebraic Latent Projection to handle hidden confounders and Natural Gradient Descent on the belief manifold for principled optimization. Experiments on synthetic and real-world benchmarks demonstrate that HOLOGRAPH provides rigorous mathematical foundations while achieving competitive performance on causal discovery tasks with 50-100 variables. Our sheaf-theoretic analysis reveals that while Identity, Transitivity, and Gluing axioms are satisfied to numerical precision (<10^{-6}), the Locality axiom fails for larger graphs, suggesting fundamental non-local coupling in latent variable projections. Code is available at [https://github.com/hyunjun1121/holograph](https://github.com/hyunjun1121/holograph).