On the Sample Complexity of Estimating Small Singular Modes
ISIT(2020)
摘要
While it is commonly believed that estimating the small singular modes for a nearly low-rank matrix requires more samples, the sample size needed is generally unclear. In this paper, we investigate this sample complexity by considering the difference between the estimation errors of estimating a matrix with or without estimating these small singular modes. Specifically, we develop a mathematical framework based on the matrix perturbation analysis to characterize the noise level of estimating small singular modes by n samples. In particular, we show that under mild assumptions on the sample noise, it requires at least n = O(η−2) samples to well estimate the singular modes with the singular value in the order of some small η. More importantly, our results are applied to the channel state estimation and Hirschfeld-Gebelein-Renyi (HGR) maximal correlation problems, from which we characterize that for how many samples, utilizing the low-rank approximation in these problems are beneficial. Finally, numerical simulations are provided to verify our results.
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关键词
channel state estimation,singular value,sample noise,estimation errors,sample complexity,low-rank matrix,singular modes
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