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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 provides a novel collection of technical and theoretical tools forstudying the geometry of singular subspaces using the two-to-infinity norm.Motivated by preliminary deterministic Procrustes analysis, we consider ageneral matrix perturbation setting in which we derive a new Procrustean matrixdecomposition. Together with flexible machinery developed for thetwo-to-infinity norm, this allows us to conduct a refined analysis of theinduced perturbation geometry with respect to the underlying singular vectorseven in the presence of singular value multiplicity. Our analysis yieldssingular vector entrywise perturbation bounds for a range of popular matrixnoise models, each of which has a meaningful associated statistical inferencetask. In addition, we demonstrate how the two-to-infinity norm is the preferrednorm in certain statistical settings. Specific applications discussed in thispaper include covariance estimation, singular subspace recovery, and multiplegraph inference.  Both our Procrustean matrix decomposition and the technical machinerydeveloped for the two-to-infinity norm may be of independent interest.

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