Heteroskedastic PCA Principal Subspace Confidence Regions and Spiked Covariance Inference with Missing Data

A distributional inference and uncertainty quantification methodology (introduced by Yuling Yan, Yuxin Chen, and Jianqing Fan in 2024) for estimating and constructing valid confidence regions for the $r$-dimensional principal subspace $\boldsymbol{U}^\star \in \mathbb{O}^{d \times r}$ and linear forms of eigenvectors of a low-rank spiked covariance matrix $\boldsymbol{\Sigma}^\star = \boldsymbol{U}^\star\boldsymbol{\Lambda}^\star\boldsymbol{U}^{\star\top} \in \mathbb{R}^{d \times d}$ from $n$ incomplete sample columns observed at rate $p$ with entrywise heteroskedastic noise variances $\sigma_{i,t}^2 = \mathrm{Var}(E_{i,t})$. Because the diagonal of the inverse-probability-weighted empirical Gram matrix $\widehat{\boldsymbol{G}} = \frac{1}{n p^2}\mathcal{P}_\Omega(\boldsymbol{X})\mathcal{P}_\Omega(\boldsymbol{X})^\top$ has severe bias $\mathbb{E}[\widehat{G}_{i,i}] = \frac{1}{p}\Sigma_{i,i}^\star + \frac{1}{n p}\sum_{t=1}^n \sigma_{i,t}^2$, the method eliminates diagonal bias either by zeroing out the diagonal $\boldsymbol{M} = \mathcal{P}_{\mathrm{off-diag}}(\widehat{\boldsymbol{G}})$ (Algorithm 1, Deflated Heteroskedastic PCA) or by iterative low-rank diagonal imputation (HeteroPCA). Theorems 1–4 establish a second-order Neumann-series expansion showing that for every row $1 \le i \le d$, the aligned subspace row error $(\boldsymbol{U}\boldsymbol{H} - \boldsymbol{U}^\star)_{i,\cdot}^\top$ is asymptotically normal $\mathcal{N}(\boldsymbol{0}, \boldsymbol{S}^{(i,\star)})$, that the plug-in covariance estimator $\widehat{\boldsymbol{S}}^{(i)}$ (Equations 3.8–3.10) yields pivotal Wald statistics $W_i = (\boldsymbol{U}\widehat{\boldsymbol{H}} - \boldsymbol{U}^\star)_{i,\cdot}(\widehat{\boldsymbol{S}}^{(i)})^{-1}(\boldsymbol{U}\widehat{\boldsymbol{H}} - \boldsymbol{U}^\star)_{i,\cdot}^\top \stackrel{\mathrm{d}}{\to} \chi_r^2$, and that linear forms $\boldsymbol{c}^\top(\boldsymbol{u}_k - \boldsymbol{u}_k^\star)$ (Theorems 5–6) and entrywise covariance estimates $\widehat{\Sigma}_{i,j} - \Sigma_{i,j}^\star$ (Theorems 7–8) achieve valid $(1-\alpha)$ Gaussian confidence intervals without requiring eigengaps among the top $r$ eigenvalues for subspace and covariance inference.