用随机路径解决高维因果推断中的概率迁移难题
Causal Schrödinger Bridges: Constrained Optimal Transport on Structural Manifolds
- 将反事实推理转化为熵正则最优传输，通过扩散过程实现跨低密度区的稳定迁移
- 在10万维系统中实现0.06的低误差重建，比传统方法快数万倍
- 适合高维因果发现与复杂系统建模，尤其对结构敏感场景有显著优势
生成建模通常依赖确定性流（ODE）寻找最小作用路径。然而，在因果干预下，这些路径容易在低密度区域（离流形）失效，导致数值不稳定和前瞻控制病态。本文提出因果薛定谔桥（CSB），将反事实推理重构为熵正则最优传输问题。利用扩散过程（SDEs），CSB 能够稳健地“隧穿”支持集不匹配区域，同时严格保持结构可接受性。我们证明了结构分解定理，表明高维全局桥可精确分解为局部鲁棒转移。该定理为高维中单体架构的瓶颈提供了原则性解决方案。我们在一个全秩因果系统（d=10^5，内在秩10^5）上验证了该方法，标准无结构MLP无法收敛（MSE ~0.31）。而通过物理实现结构分解，CSB 在单个GPU上仅用73.73秒即达到高保真度传输（MSE ~0.06），相比之下，结构无关的O(d^3)基线估计需超过6年。结果表明，CSB通过结构智能打破了维度诅咒，为10^5节点系统的高风险因果发现提供了可扩展基础。
原文摘要 · Abstract (English)
Generative modeling typically seeks the path of least action via deterministic flows (ODE). While effective for in-distribution tasks, we argue that these deterministic paths become brittle under causal interventions, which often require transporting probability mass across low-density regions ("off-manifold") where the vector field is ill-defined. This leads to numerical instability and the pathology of anticipatory control. In this work, we introduce the Causal Schrodinger Bridge (CSB), a framework that reformulates counterfactual inference as Entropic Optimal Transport. By leveraging diffusion processes (SDEs), CSB enables probability mass to robustly "tunnel" through support mismatches while strictly enforcing structural admissibility. We prove the Structural Decomposition Theorem, showing that the global high-dimensional bridge factorizes exactly into local, robust transitions. This theorem provides a principled resolution to the Information Bottleneck that plagues monolithic architectures in high dimensions. We empirically validate CSB on a full-rank causal system (d=10^5, intrinsic rank 10^5), where standard structure-blind MLPs fail to converge (MSE ~0.31). By physically implementing the structural decomposition, CSB achieves high-fidelity transport (MSE ~0.06) in just 73.73 seconds on a single GPU. This stands in stark contrast to structure-agnostic O(d^3) baselines, estimated to require over 6 years. Our results demonstrate that CSB breaks the Curse of Dimensionality through structural intelligence, offering a scalable foundation for high-stakes causal discovery in 10^5-node systems. Code is available at: https://github.com/cochran1/causal-schrodinger-bridge
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