We develop a distributionally robust formulation of principal component analysis that minimizes worst-case reconstruction risk over distributions lying within a Wasserstein neighborhood of the empirical measure. The Wasserstein neighborhood, viewed as an ambiguity set of distributions, is adaptively calibrated through a transport matrix $G$ to capture heterogeneous uncertainty across dimensions. The homogeneous case, in which G is a scalar multiple of the identity matrix, recovers classical PCA. Under a general transport matrix G, we derive a dual characterization of the associated minimax optimization problem and introduce a tractable surrogate objective function consisting of the square-root empirical reconstruction error plus a geometry-dependent residual exposure penalty. The exact and surrogate estimators are shown to be consistent for the population PCA subspace and asymptotically equivalent at the projector level. The transport geometry is allowed to be data adaptive, while the Wasserstein radius is calibrated via robust Wasserstein profile inference, yielding a data-driven radius of order $n^{-1/2}$. Comprehensive theoretical guarantees are established, including consistency and local Grassmannian asymptotics exhibiting an explicit Wasserstein-induced drift determined by the limiting transport geometry and calibration level. Numerical experiments and a real-data application demonstrate that the proposed method can substantially improve finite-sample out-of-sample performance under structured covariance shifts, moderate contamination, and certain same-distribution regimes.

翻译：我们提出了一种基于分布鲁棒性的主成分分析框架，该框架通过最小化经验测度Wasserstein邻域内分布的最坏重构风险来实现。我们将Wasserstein邻域视为一个分布模糊集，并通过传输矩阵$G$对其自适应校准，以捕捉维度间的异质性不确定性。当$G$为单位矩阵的标量倍时，该同质情形可恢复经典主成分分析。在一般传输矩阵$G$下，我们推导了对应极小极大优化问题的对偶表征，并引入一个可解的替代目标函数，该函数由平方根经验重构误差与依赖几何结构的残差暴露惩罚项组成。我们证明精确估计量与替代估计量对于总体主成分子空间具有相合性，且在投影算子层面渐进等价。传输几何结构可进行数据自适应调整，而Wasserstein半径通过鲁棒Wasserstein轮廓推断进行校准，得到阶数为$n^{-1/2}$的数据驱动半径。本文建立了完整的理论保证，包括相合性及展现显式Wasserstein诱导漂移的局部Grassmann渐近性质，该漂移由极限传输几何结构与校准水平决定。数值实验与真实数据应用表明，在结构化协方差偏移、中等污染及部分同分布场景下，所提方法能够显著提升有限样本的样本外性能。