Hessian 秩约束用于非线性潜变量模型的结构学习

中文摘要 AI 辅助

从观测数据中揭示潜变量及其因果关系是一个基础但具有挑战性的问题。现有方法通常依赖限制性假设，如线性关系或可逆混合函数。为了更好地在一般非线性混合过程下解决此问题，我们提出了一种称为交叉 Hessian 秩约束（HRC）的条件，该条件作为非线性潜变量因果发现的基于秩的原始工具。特别地，我们证明了在非线性情况下，观测数据对数密度的交叉 Hessian 会产生一种基于秩的性质，揭示有关潜变量的信息，并在线性高斯情况下简化为 Tetrad 约束。更具体地，当两组观测变量被一组较低维潜变量 d-分离时，在条件对数密度导数的温和仿射导数假设下，该交叉 Hessian 的秩等于潜变量的维度。当噪声水平较低或相关非线性适中时，该假设可以自然满足。作为下游应用，我们在纯单因子测量设置中实例化 HRC，用于定位潜变量并恢复其因果结构直至马尔可夫等价。在合成和真实数据集上的实验结果支持理论主张。

英文摘要

Uncovering latent variables and their causal relations from observed data is a fundamental yet challenging problem. Existing methods often rely on restrictive assumptions, such as linear relations or invertible mixing functions. To better address this problem under general nonlinear mixing procedures, we propose a condition called the cross-Hessian Rank Constraint (HRC), which serves as a primitive rank-based tool for nonlinear latent causal discovery. In particular, we show that a rank-based property arises from the cross-Hessian of the observed-data log-density in the nonlinear case, revealing information about the latent variables, and reduces to the Tetrad constraints in the linear Gaussian case. More specifically, when two groups of observed variables are d-separated by a set of lower-dimensional latent variables, the rank of this cross-Hessian is equal to the dimension of the latent variables, under a mild affine derivative assumption on the conditional log-density derivatives. This assumption can be naturally satisfied when the noise level is low or the relevant nonlinearity is moderate. As a downstream application, we instantiate HRC in the pure one-factor measurement setting for locating latent variables and recovering their causal structure up to Markov equivalence. Experimental results on synthetic and real-world datasets support the theoretical claims.

发表机构

• Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
• Carnegie Mellon University（卡内基梅隆大学）
• Guangdong University of Technology（广东工业大学）
• Beijing Technology and Business University（北京工商大学）
• University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
• Johns Hopkins University（约翰斯·霍普金斯大学）

机构由 AI 辅助整理，请以论文原文为准。