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Activity Number: 547 - Annals of Statistics Special Invited Session: Selected Papers
Type: Invited
Date/Time: Wednesday, July 31, 2019 : 2:00 PM to 3:50 PM
Sponsor: IMS
Abstract #300111 Presentation
Title: The Two-to-Infinity Norm and Singular Subspace Geometry
Author(s): Carey E Priebe* and Minh Tang and Joshua Cape
Companies: Johns Hopkins University and Johns Hopkins University and Johns Hopkins University
Keywords: singular value decomposition; principal component analysis; eigenvector perturbation; spectral methods; Procrustes analysis; high-dimensional statistics

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 us to conduct 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 singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings.

Authors who are presenting talks have a * after their name.

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