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Spectral methods have myriad applications in high-dimensional statistics anddata science, and while previous works have primarily focused on $\ell_2$ or$\ell_{2,\infty}$ eigenvector and singular vector perturbation theory, in manysettings these analyses fall short of providing the fine-grained guaranteesrequired for various inferential tasks. In this paper we study statisticalinference for linear functions of eigenvectors and principal components with aparticular emphasis on the setting where gaps between eigenvalues may beextremely small relative to the corresponding spiked eigenvalue, a regime whichhas been oft-neglected in the literature. It has been previously establishedthat linear functions of eigenvectors and principal components incur anon-negligible bias, so in this work we provide Berry-Esseen bounds forempirical linear forms and their debiased counterparts respectively in thematrix denoising model and the spiked principal component analysis model, bothunder Gaussian noise. Next, we propose data-driven estimators for theappropriate bias and variance quantities resulting in approximately validconfidence intervals, and we demonstrate our theoretical results throughnumerical simulations. We further apply our results to obtain distributionaltheory and confidence intervals for eigenvector entries, for which debiasing isnot necessary. Crucially, our proposed confidence intervals and bias-correctionprocedures can all be computed directly from data without sample-splitting andare asymptotically valid under minimal assumptions on the eigengap and signalstrength. Furthermore, our Berry-Esseen bounds clearly reflect the effects ofboth signal strength and eigenvalue closeness on the estimation and inferencetasks.

Distributional Theory and Statistical Inference for Linear Functions of Eigenvectors with Small Eigengaps