用于恢复潜在几何的Wasserstein马氏距离
Wasserstein Mahalanobis Distances for Recovering Latent Geometry
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中文总结 AI 辅助
该研究提出Wasserstein马氏距离，将马氏距离从向量数据扩展到概率测度，可在Wasserstein空间中恢复潜在几何，其数值实验验证了理论预测的有效性。
中文摘要 AI 辅助
马氏距离是适用于多元数据的、基于协方差调整的基础度量，在从非线性观测中恢复潜在几何方面发挥核心作用。我们将这一原理从向量值数据扩展到概率测度，提出了Wasserstein马氏距离。我们的构造用最优输运位移场替代欧氏位移向量，用Wasserstein切空间上定义的协方差算子替代局部协方差矩阵。我们证明该构造继承了非线性独立成分分析所基于的几何恢复性质。特别地，对于经光滑非线性推前变换的具有共同协方差的高斯测度，所提出的Wasserstein马氏距离可近似变换后潜在均值之间的经典马氏距离；对于仿射变换，这种对应关系是精确的，对于一般光滑变换，其对应关系在可控高阶误差项范围内成立。这些结果建立了经典马氏几何的分布值类比，为直接在Wasserstein空间中进行协方差调整的学习提供了理论支持。数值实验验证了理论预测，证明能准确恢复潜在几何结构。
英文摘要
The Mahalanobis distance is a fundamental covariance-adapted metric for multivariate data and plays a central role in recovering latent geometry from nonlinear observations. We extend this principle from vector-valued data to probability measures by introducing a Wasserstein Mahalanobis distance. Our construction replaces Euclidean displacement vectors with optimal transport displacement fields and local covariance matrices with covariance operators defined on Wasserstein tangent spaces. We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis. In particular, for Gaussian measures with common covariance transformed by a smooth nonlinear pushforward, the proposed Wasserstein Mahalanobis distance approximates the classical Mahalanobis distance between the transformed latent means. The correspondence is exact for affine transformations and holds up to controlled higher-order error terms for general smooth transformations. These results establish a distribution-valued analog of classical Mahalanobis geometry and provide theoretical support for covariance-adapted learning directly in Wasserstein space. Numerical experiments confirm the theoretical predictions and demonstrate accurate recovery of latent geometric structure.
