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Title: Inference for heteroskedastic PCA with missing data Authors:  Yuling Yan - Princeton University (United States) [presenting]
Yuxin Chen - University of Pennsylvania (United States)
Jianqing Fan - Princeton University (United States)
Abstract: The purpose is to study how to construct confidence regions for principal component analysis (PCA) in high dimension, a vastly under-explored problem. While computing measures of uncertainty for nonlinear/nonconvex estimators is in general difficult in high dimensions, 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. 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 random noise, without requiring prior knowledge about the noise levels and noise distributions.