We study the statistical decision process of detecting the low-rank signal from various signal-plus-noise type data matrices, known as the spiked random matrix models. We first show that the principal component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non-Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked random matrices, which generalize the Baik-Ben Arous-P\'{e}ch\'{e} (BBP) transition. We also prove the central limit theorem for the linear spectral statistics for the spiked random matrices and propose a hypothesis test based on it, which does not depend on the distribution of the signal or the noise. When the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix with additive noise. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.
翻译:我们研究从各种信号加噪音类型的数据矩阵中检测低声信号的统计决策程序,即加压随机矩阵模型。我们首先显示,如果噪音不是古日文,主要组成部分分析可以通过输入预变数据矩阵来改进。如果噪音是非古日文,我们一般地将Baik-Ben Arous-P\{e\{e}(BBP)转型,我们发现,在使用一级信号的奇特随机矩阵模型模型中,已知结果与一级信号或噪音的分布无关。作为中间步骤,我们发现,对加压随机矩阵的极端电子值的急剧阶段过渡阈值,它一般地将Baik-Ben Araus-P\{e}{e}(BBPP)转换。我们还证明了对加压随机矩阵的线光谱统计的中心限制,并在此基础上提出假设测试,这不取决于信号的分布或噪音。当噪音是非古日文时,试验可以随着向带有添加噪音的数据矩阵的数据矩阵的入式转换而改进。我们还引入一种算法,在事先不知道信号的状态时估计信号的等级。