  This paper studies how to construct confidence regions for principal
component analysis (PCA) in high dimension, a problem that has been vastly
under-explored. 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 novel approach to performing valid inference on the principal
subspace under a spiked covariance model with missing data, on the basis of an
estimator called HeteroPCA (Zhang et al., 2022). We develop non-asymptotic
distributional guarantees for HeteroPCA, 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. Our inference
procedures are fully data-driven and adaptive to heteroskedastic random noise,
without requiring prior knowledge about the noise levels.
