Analysis of singular subspaces under random perturbations
arXiv:2403.09170v3 Announce Type: replace-cross
Abstract: We present a comprehensive analysis of singular vector and singular subspace perturbations in the signal-plus-noise matrix model with random Gaussian noise. Assuming a low-rank signal matrix, we extend the Davis-Kahan-Wedin theorem in a fully generalized manner, applicable to any unitarily invariant matrix norm, building on previous results by O’Rourke, Vu, and the author. Our analysis provides fine-grained insights, including $ell_infty$ bounds for singular vectors, $ell_{2, infty}$ bounds for singular subspaces, and results for linear and bilinear functions of singular vectors. Additionally, we derive $ell_{2,infty}$ bounds on perturbed singular vectors, taking into account the weighting by their corresponding singular values. Finally, we explore practical implications of these results in the Gaussian mixture model and the submatrix localization problem.
