Throughout this paper, we assume that the entries of W are subgaussian.

Throughout this paper, we assume that the entries of W are subgaussian.

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 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 $\textsf{HeteroPCA}$ (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 $\textsf{HeteroPCA}$, 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.

Independent component analysis (ICA) has been a popular dimension reduction tool in statistical machine learning and signal processing. In this paper, we present a convergence analysis for an online tensorial ICA algorithm, by viewing the problem as a nonconvex stochastic approximation problem. For estimating one component, we provide a dynamics-based analysis to prove that our online tensorial ICA algorithm with a specific choice of stepsize achieves a sharp finite-sample error bound. In particular, under a mild assumption on the data-generating distribution and a scaling condition such that $d^4 / T$ is sufficiently small up to a polylogarithmic factor of data dimension $d$ and sample size $T$, a sharp finite-sample error bound of $\tilde O(\sqrt{d / T})$ can be obtained. As a by-product, we also design an online tensorial ICA algorithm that estimates multiple independent components in parallel, achieving desirable finite-sample error bound for each independent component estimator.