Abstract:
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In many probability, statistics and machine learning applications, singular vector perturbation bound, which describes how the singular vectors change after possible perturbations to the matrices, has been widely considered. The prominent Wedin (1972)'s sine-theta law addressed this question by providing a uniform perturbation bound on both left and right singular vectors. To achieve respective better rate on one side of singular vector perturbation, in this article we propose a new perturbation bound on unilateral singular vectors. Both upper bound and lower bound results are developed for perturbation under both spectral and Frobenius sine-theta norms. The proposed technical tool is further applied to a number of problems including matrix denoising, canonical correlation analysis, where we are able to provide better results comparing to the literature. Some related problems are also briefly discussed.
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