Inference for Heteroskedastic PCA with Missing Data

• Activity Number: 188 - SLDS Student Paper Awards
• Type: Topic Contributed
• Date/Time: Monday, August 8, 2022 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Yuling Yan* and Yuxin Chen and Jianqing Fan
• Companies: Princeton University and University of Pennsylvania and Princeton University
• Keywords: principal component analysis; confidence regions; missing data; uncertainty quantification; heteroskedastic data; subspace estimation

Abstract

This paper studies how to construct confidence regions for principal component analysis (PCA) in high dimension. While computing measures of uncertainty for nonlinear/nonconvex estimators is in general difficult in high dimension, the challenge is further compounded by the prevalent presence of missing data and heteroskedastic noise. We propose a suite of solutions to perform valid inference on the principal subspace based on two estimators: a vanilla SVD-based approach, and a more refined iterative scheme called HeteroPCA (Zhang et al., 2018). We develop non-asymptotic distributional guarantees for both estimators, and demonstrate how these can be invoked to compute both confidence regions for the principal subspace and entrywise confidence intervals for the spiked covariance matrix. Particularly worth highlighting is the inference procedure built on top of HeteroPCA, which is not only valid but also statistically efficient for broader scenarios (e.g., it covers a wider range of missing rates and signal-to-noise ratios). Our solutions are fully data-driven and adaptive to heteroskedastic noise, without requiring prior knowledge about the noise levels and noise distributions.