The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

摘要

The singular value matrix decomposition plays a ubiquitous role throughoutstatistics and related fields. Myriad applications including clustering,classification, and dimensionality reduction involve studying and exploitingthe geometric structure of singular values and singular vectors. This paper contributes to the literature by providing a novel collection oftechnical and theoretical tools for studying the geometry of singular subspacesusing the $2ightarrowinfty$ norm. Motivated by preliminary deterministicProcrustes analysis, we consider a general matrix perturbation setting in whichwe derive a new Procrustean matrix decomposition. Together with flexiblemachinery developed for the $2ightarrowinfty$ norm, this allows us toconduct a refined analysis of the induced perturbation geometry with respect tothe underlying singular vectors even in the presence of singular valuemultiplicity. Our analysis yields perturbation bounds for a range of popularmatrix noise models, each of which has a meaningful associated statisticalinference task. We discuss how the $2ightarrowinfty$ norm is arguably thepreferred norm in certain statistical settings. Specific applications discussedin this paper include the problem of covariance matrix estimation, singularsubspace recovery, and multiple graph inference. Both our novel Procrustean matrix decomposition and the technical machinerydeveloped for the $2ightarrowinfty$ norm may be of independent interest.