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

Understand

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.