Title: The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics,
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.
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.