We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where the noise matrix is allowed to have dependence within rows and heteroskedasticity between them. We prove finite-sample $\ell_{2,\infty}$ bounds and a Berry-Esseen theorem for the individual entries of the estimator, and we apply these results to high-dimensional mixture models. Our Berry-Esseen theorem clearly shows the geometric relationship between the signal matrix, the covariance structure of the noise, and the distribution of the errors in the singular vector estimation task. These results are illustrated in numerical simulations. Unlike previous results of this type, which rely on assumptions of gaussianity or independence between the entries of the additive noise, handling the dependence between entries in the proofs of these results requires careful leave-one-out analysis and conditioning arguments. Our results depend only on the signal-to-noise ratio, the sample size, and the spectral properties of the signal matrix.
翻译:我们为高维低位矩阵的单向矢量建议一个估计器, 高维低位矩阵的单向矢量被添加的亚双星噪音腐蚀, 噪音矩阵允许在行内产生依赖性和它们之间的偏移性。 我们用数字模拟来说明这些结果, 与先前的这种类型的结果不同, 这种结果依赖于添加噪声条目的假设或独立性, 处理这些结果的证明中条目之间的依赖性需要谨慎的请假单向外分析和调节参数。 我们的Berry- Esseen 理论清楚显示了信号矩阵、 噪音的变量以及单向矢量估计任务中错误的分布之间的几何关系。 这些结果与先前的模拟不同, 取决于添加噪声的条目之间的假设或独立性, 处理这些结果的证明中条目之间的依赖性需要谨慎的请假分析, 和调节参数。 我们的结果只取决于信号对声比、 样本大小和信号矩阵的谱属性。