通过与高斯分布的 Wasserstein 距离实现无对比独立成分分析和因果推断
Contrast-Free ICA and Causal Inference via Wasserstein Distances to the Gaussian
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中文总结 AI 辅助
研究利用到高斯分布的 Wasserstein 距离作非高斯性准则，用于线性 ICA 和 LiNGAM 因果推断。基于源与线性组合的 Wasserstein 距离不等式，总体可精确识别 ICA 矩阵与因果顺序，还给出估计器及求解器，实验展现良好性能并开源实现。
中文摘要 AI 辅助
我们研究了到标准高斯分布的平方 2-Wasserstein 距离作为非高斯性准则，并将其用于线性独立成分分析（ICA）和线性非高斯无环模型（LiNGAM）中的因果推断。该分析依赖于独立标准化源的 Wasserstein 非高斯性与其线性组合之间的严格不等式。当至多一个源是高斯分布时，任何涉及至少两个源的单位范数线性组合的平方 Wasserstein 距离都比源距离的相应加权和严格小。在总体层面，这可精确识别 ICA 解混矩阵（至多符号置换），并通过最小二乘残差对因果顺序进行类似刻画。接着定义了经验插件估计器，并在有限矩假设下证明了无分布一致收敛界，然后详细介绍了三种实用求解器：用于 ICA 的 Picard 风格正交优化器、用于因果顺序搜索的穷举动态规划以及贪婪顺序搜索变体。通过实验，我们展示了这两个任务的竞争性能，并提供了源分离和因果推断的开源实现。
英文摘要
We study the squared $2$-Wasserstein distance to the standard Gaussian as a non-Gaussianity criterion and use it for linear Independent Component Analysis (ICA) and causal discovery in Linear Non-Gaussian Acyclic Models (LiNGAM). Unlike commonly used parametric contrasts and approximations of information-theoretic quantities, this criterion requires no distributional regularity beyond finite second moments, involves neither approximation nor tuning parameters, and can be computed exactly and efficiently from empirical order statistics. Our analysis relies on a strict subadditivity property of the $2$-Wasserstein distance to the Gaussian. At the population level, we prove exact identification of the ICA unmixing matrix, up to signed permutation, and give an analogous characterization of causal orders through sequential least-squares residuals. We then define empirical plug-in estimators and prove distribution-free uniform convergence under finite-moment assumptions, before detailing three practical solvers: a Picard-style orthogonal optimizer for ICA, an exhaustive dynamic program for causal order search, and a greedy order search variant. Empirically, we demonstrate competitive performance for both tasks and provide open-source implementations for source separation and causal discovery.
