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

## TL;DR

This paper introduces new theoretical tools using the two-to-infinity norm to analyze the geometry of singular subspaces, with applications in high-dimensional statistics such as covariance estimation and graph inference.

## Contribution

It develops a novel Procrustean matrix decomposition and refined perturbation bounds for singular vectors, especially in cases with singular value multiplicity.

## Key findings

- Derived singular vector entrywise perturbation bounds for various noise models
- Showed the two-to-infinity norm's effectiveness in statistical inference tasks
- Provided applications in covariance estimation and graph inference

## 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.