Matrices with Gaussian noise: optimal estimates for singular subspace perturbation

2018/03/02 by Sean O’Rourke, Van Vu, O'Rourke, Sean +3 · 6 citations
Mathematics · #Random Matrices and Applications #Point processes and geometric inequalities #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.1803.00679

Abstract

The Davis-Kahan-Wedin sin Θ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin sin Θ theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis-Kahan-Wedin sin Θ theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.

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