Tackling Small Eigen-Gaps: Fine-Grained Eigenvector Estimation and Inference Under Heteroscedastic Noise

This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: 1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; 2) how to perform estimation and inference on linear functionals of an eigenvector—a ...