Optimization-Free Topological Sort for Causal Discovery via the Schur Complement of Score Jacobians

TL;DR

This paper introduces a novel, optimization-free method for causal discovery that leverages the geometric signature of the score function and the Schur complement, enabling scalable analysis of high-dimensional non-linear graphs.

Contribution

It proposes the Score-Schur Topological Sort (SSTS), a new algorithm that extracts causal order directly from generative models without non-convex optimization, using algebraic operations.

Findings

SSTS enables causal analysis on graphs with up to 1000 nodes.

The method reduces graph extraction to matrix operations, improving scalability.

Structural fidelity is bounded by finite-sample score estimation variance.

Abstract

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.