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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.25295 |
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| _version_ | 1866908997630558208 |
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| author | Wu, Rui Xie, Hong |
| author_facet | Wu, Rui Xie, Hong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25295 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimization-Free Topological Sort for Causal Discovery via the Schur Complement of Score Jacobians Wu, Rui Xie, Hong Machine Learning 68T05, 62H22 I.2.6; G.3 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. |
| title | Optimization-Free Topological Sort for Causal Discovery via the Schur Complement of Score Jacobians |
| topic | Machine Learning 68T05, 62H22 I.2.6; G.3 |
| url | https://arxiv.org/abs/2604.25295 |