Singular value

Mathematical analysis

Upper and lower bounds

Combinatorics

Physics

Perturbation (astronomy)

Quantum mechanics

Singular perturbation

Pure mathematics

Moore–Penrose pseudoinverse

Mathematics

Discrete mathematics

Linear subspace

Eigenvalues and eigenvectors

Inverse

Geometry

The Annals of Statistics

Perturbation bounds for singular spaces, in particular Wedin’s $\mathop{\mathrm{sin}}\nolimits \Theta$ 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 $\mathop{\mathrm{sin}}\nolimits \Theta$ 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 u...

Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics