Matrices with Gaussian noise: optimal estimates for singular subspace perturbation

Sean O'Rourke; Van Vu; Ke Wang

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arXiv · 1803.00679

Matrices with Gaussian noise: optimal estimates for singular subspace perturbation

Sean O'Rourke·Van Vu·Ke Wang

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Abstract

The Davis-Kahan-Wedin $\sin \Theta$ 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 $\sin \Theta$ 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 $\sin \Theta$ 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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BibTeX RIS

Sean O'Rourke, Van Vu, Ke Wang. 2018-03-02. Matrices with Gaussian noise: optimal estimates for singular subspace perturbation. https://arxiv.org/abs/1803.00679

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