Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence

• Activity Number: 560 - Latent Space Modeling and Dimensionality Reduction
• Type: Contributed
• Date/Time: Thursday, August 11, 2022 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Joshua Agterberg* and Zachary Lubberts and Carey E Priebe
• Companies: Johns Hopkins University and Johns Hopkins University and Johns Hopkins University
• Keywords: Singular Value Decomposition; Heteroskedasticity; Matrix Analysis; Eigenvectors; Entrywise Perturbation; Data Science

Abstract

We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where the noise matrix is allowed to have dependence within rows and heteroskedasticity between them. We prove finite-sample row-wise perturbation bounds and a Berry-Esseen theorem for the individual entries of the estimator, and we apply these results to high-dimensional mixture models. Our Berry-Esseen theorem clearly shows the geometric relationship between the signal matrix, the covariance structure of the noise, and the distribution of the errors in the singular vector estimation task. These results are illustrated in numerical simulations. Unlike previous results of this type, which rely on assumptions of gaussianity or independence between the entries of the additive noise, handling the dependence between entries in the proofs of these results requires careful leave-one-out analysis and conditioning arguments. Our results depend only on the signal-to-noise ratio, the sample size, and the spectral properties of the signal matrix.