Distributionally Robust PCA with Data-Adaptive Wasserstein Geometry

2026-06-09

AI Summary

This work addresses the instability of reconstruction risk in principal component analysis (PCA) under distributional shifts and data contamination by proposing a distributionally robust PCA framework. The method minimizes the worst-case reconstruction risk over a Wasserstein ambiguity set, incorporating a data-adaptive transport matrix to capture heterogeneous uncertainty across dimensions and leveraging robust Wasserstein profile inference to automatically calibrate the neighborhood radius. Theoretically, the approach establishes consistency of the estimated subspace and asymptotic equivalence on the local Grassmann manifold. Empirical results demonstrate that the proposed method significantly improves out-of-sample reconstruction performance under finite samples, particularly in settings involving structured covariance shifts, moderate contamination, and even standard i.i.d. conditions.

Abstract

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.

Problem

• Distributionally Robust PCA
• Wasserstein Geometry
• Reconstruction Risk
• Covariance Shift
• Out-of-Sample Performance

Innovation

• Distributionally Robust PCA
• Data-Adaptive Wasserstein Geometry
• Transport Matrix
• Robust Wasserstein Profile Inference
• Residual Exposure Penalty