A1628
Title: Distributionally robust PCA with data-adaptive Wasserstein geometry
Authors: Yanrong Yang - The Australian National University (Australia)
Chuang Xu - Australian National University (Australia) [presenting]
Abstract: A distributionally robust formulation of PCA is developed that minimizes worst-case reconstruction risk over distributions lying within a Wasserstein neighborhood of the empirical measure. The Wasserstein neighborhood 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, a dual characterization of the associated minimax optimization problem is derived and a tractable surrogate objective function is introduced 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 $n^{-1/2}$. 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 method can improve out-of-sample performance under structured covariance shifts, moderate contamination, and certain same-distribution regimes.
