Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics

TL;DR

This paper develops rate-optimal perturbation bounds for singular subspaces, providing separate bounds for left and right spaces, with applications to high-dimensional statistics and machine learning tasks.

Contribution

It introduces the first bounds that distinguish between left and right singular spaces, establishing their rate-optimality and broad applicability.

Findings

Separate bounds for left and right singular spaces are established.

Matching upper and lower bounds demonstrate rate-optimality.

Applications include matrix denoising, clustering, and CCA.

Abstract

Perturbation bounds for singular spaces, in particular Wedin's $\sin \Theta$ theorem, are a fundamental tool in many fields including high-dimensional statistics, machine learning, and applied mathematics. In this paper, we establish separate perturbation bounds, measured in both spectral and Frobenius $\sin \Theta$ distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given. The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering, and canonical correlation analysis (CCA). In particular, separate matching upper and lower bounds are obtained for estimating the left and right singular spaces. To the best of our knowledge, this is the first result that gives different optimal rates for the left and right singular spaces under the same perturbation. In addition to these problems, applications to other high-dimensional problems such as community detection in bipartite networks, multidimensional scaling, and cross-covariance matrix estimation are also discussed.