Open Access
October 2019 The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics
Joshua Cape, Minh Tang, Carey E. Priebe
Ann. Statist. 47(5): 2405-2439 (October 2019). DOI: 10.1214/18-AOS1752


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.

This paper provides 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. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference.

Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.


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Joshua Cape. Minh Tang. Carey E. Priebe. "The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics." Ann. Statist. 47 (5) 2405 - 2439, October 2019.


Received: 1 May 2017; Revised: 1 March 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114917
MathSciNet: MR3988761
Digital Object Identifier: 10.1214/18-AOS1752

Primary: 62H12 , 62H25
Secondary: 62H30

Keywords: eigenvector perturbation , High-dimensional statistics , Principal Component Analysis , Procrustes analysis , Singular value decomposition , spectral methods

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 5 • October 2019
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