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Center for Nonlinear Studies |
The power method is a simple and efficient algorithm for finding the top $k$ singular vectors of any input matrix. In practice, a noise matrix could be added to the input matrix at each iteration of the power method, and the convergence behavior of the algorithm is not hard to guarantee. In this paper, we address problems of the noisy power method. The name noisy power method is borrowed from [Hardt et al 2014]. The convergence behavior of the noisy power method is understood only for the cases when the noise level (the spectral norm of noise matrices) is bellow a threshold, and there are many open questions as stated in [Hardt et al 2014]. Moreover, the noisy power method cannot extract the exact top $k$ singular vectors because of the noise matrices. Our contributions are three folds: {\em i)} we provide a different approach to analyze the noisy power method that will help understanding the convergence behavior of the noisy power method, {\em ii)} we propose a simple add-on algorithm that makes the output converge to the exact top $k$ singular vectors, {\em iii)} we also provide a negative example where the noisy power method performs very badly when the noise level is equal to the spectral gap, which is a counter example for Conjecture 1.2 of [Hardt et al 2014]​.​
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