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This paper introduces Score-Schur Topological Sort (SSTS), a novel algorithm for causal discovery that bypasses non-convex acyclicity penalties by extracting topological order directly from unconstrained generative models. SSTS leverages the Schur complement of the Score-Jacobian Information Matrix (SJIM) to translate the acyclicity constraint into an algebraic procedure, enabling efficient causal structure learning. Experiments on non-linear graphs up to d=1000 demonstrate that SSTS shifts the bottleneck from optimization to statistical score estimation, revealing that structural fidelity is bounded by the finite-sample estimation variance of the global score geometry.
Forget struggling with non-convex optimization for causal discovery: a new algorithm extracts causal order directly from score functions with simple matrix operations.
Continuous causal discovery typically couples representation learning with structural optimization via non-convex acyclicity penalties, which subjects solvers to local optima and restricts scalability in high-dimensional regimes. We propose a decoupled paradigm that shifts the causal discovery bottleneck from non-convex optimization to statistical score estimation. We introduce the Score-Schur Topological Sort (SSTS), an algorithm that extracts topological order directly from unconstrained generative models, bypassing constrained structure optimization. We establish that the causal hierarchy leaves a geometric signature within the score function: iterative graph marginalization is mathematically equivalent to computing the Schur complement of the Score-Jacobian Information Matrix (SJIM) under linear conditions. This translates the acyclicity constraint into an algebraic procedure with a dominant cost of O(d^3) operations. For non-linear systems, we formulate the expectation gap of Schur marginalization and introduce Block-SSTS to compress extraction depth, bounding structural error. Empirically, SSTS allows causal structural analysis on non-linear graphs up to d=1000. At this scale, our framework indicates that once the non-convex optimization bottleneck is mathematically bypassed, the structural fidelity of continuous causal discovery is bounded by the finite-sample estimation variance of the global score geometry. By reducing graph extraction to matrix operations, this work reframes scalable causal discovery from a constrained optimization problem to a statistical estimation challenge.
