Statistical Inference for Linear Functions of Eigenvectors with Small Eigengaps

Spectral methods have myriad applications in high-dimensional statistics and data science, and while previous works have primarily focused on $\ell_2$ or $\ell_{2,\infty}$ eigenvector and singular vector perturbation theory, in many settings these analyses fall short of providing the fine-grained guarantees required for various inferential tasks. In this paper we study statistical inference for linear functions of eigenvectors and principal components with a particular emphasis on the setting where gaps between eigenvalues may be extremely small relative to the corresponding spiked eigenvalue, a regime which has been oft-neglected in the literature. First, we prove the approximate Gaussianity for debiased linear forms in the matrix denoising model and the spiked principal component analysis model, both under Gaussian noise. Based on this limiting behavior, we propose estimators for the appropriate bias and variance quantities resulting in approximately valid confidence intervals. We then investigate the optimality of these confidence intervals and show that their widths are minimax optimal up to constant factors. Of note, our proposed confidence intervals can be computed directly from data without the need for any sample-splitting.