基于语义嵌入原理，从无干预数据中学习因果抽象映射。
Causal Abstraction Learning based on the Semantic Embedding Principle
- 通过语义嵌入原则，将高层与低层因果模型映射关系建模为流形上的几何问题。
- 在合成数据和真实脑数据上均成功学习线性因果抽象，即使缺乏结构先验。
- 适用于缺乏干预数据、样本错位等现实场景，适合因果推理与多尺度建模研究者。
结构化因果模型（SCMs）允许我们在多个分辨率层次上分析复杂系统。因果抽象（CA）框架形式化了高层与低层SCMs之间的映射关系。本文在一种具有挑战性且现实的设定下研究CA学习：无法获取SCMs，无干预数据，且样本数据不对齐。核心思想是语义嵌入原则，即高层分布位于低层分布的子空间中。该原则自然地将线性CA与施蒂费尔流形的几何结构关联起来。我们提出基于范畴论的SCM方法，通过寻找低层与高层概率测度间的态射来学习CA，遵循语义嵌入原则。由此构建了一般性CA学习问题。作为应用，针对线性CA求解该问题，假设高斯分布并以KL散度为目标函数。由于优化问题非凸，我们设计了三种基于黎曼优化范式的算法。实验表明，所提方法在合成数据和真实脑数据上均表现良好，且对因果抽象结构的先验信息要求灵活。
原文摘要 · Abstract (English)
Structural causal models (SCMs) allow us to investigate complex systems at multiple levels of resolution. The causal abstraction (CA) framework formalizes the mapping between high- and low-level SCMs. We address CA learning in a challenging and realistic setting, where SCMs are inaccessible, interventional data is unavailable, and sample data is misaligned. A key principle of our framework is semantic embedding, formalized as the high-level distribution lying on a subspace of the low-level one. This principle naturally links linear CA to the geometry of the Stiefel manifold. We present a category-theoretic approach to SCMs that enables the learning of a CA by finding a morphism between the low- and high-level probability measures, adhering to the semantic embedding principle. Consequently, we formulate a general CA learning problem. As an application, we solve the latter problem for linear CA; considering Gaussian measures and the Kullback-Leibler divergence as an objective. Given the nonconvexity of the learning task, we develop three algorithms building upon existing paradigms for Riemannian optimization. We demonstrate that the proposed methods succeed on both synthetic and real-world brain data with different degrees of prior information about the structure of CA.
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