The Two-To-Infinity Norm and Singular Subspace Geometry with Applications to High-Dimensional Statistics

• Activity Number: 594 - Methods for Analysis of High-Dimensional Data
• Type: Contributed
• Date/Time: Wednesday, August 1, 2018 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Joshua Cape* and Minh Tang and Carey E Priebe
• Companies: Johns Hopkins; Dept. of Applied Math and Statistics and Johns Hopkins University and Johns Hopkins University
• Keywords: high-dimensional statistics; dimension reduction; singular value decomposition; spectral methods; perturbation theory

Abstract

The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. We provide a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows for a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. Specific applications include the problem of covariance estimation, singular subspace recovery, and multiple graph inference.