Characterizing the exact asymptotic distributions of high-dimensional eigenvectors for large structured random matrices poses important challenges yet can provide useful insights into a range of applications. To this end, in this paper we introduce a general framework of asymptotic theory of eigenvectors (ATE) for large structured symmetric random matrices with heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise, where the mean matrix is assumed to have the low-rank structure. Under some mild regularity conditions, we provide the asymptotic expansions for the spiked eigenvalues and show that they are asymptotically normal after some normalization. For the spiked eigenvectors, we establish novel asymptotic expansions for the general linear combination and further show that it is asymptotically normal after some normalization, where the weight vector can be arbitrary. We also provide a more general asymptotic theory for the spiked eigenvectors using the bilinear form. Simulation studies verify the validity of our new theoretical results. Our family of models encompasses many popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis, and our general theory can be exploited for statistical inference in these large-scale applications.
翻译:用于大型结构随机矩阵的高维成份分布的精确度分布,是一个重要的挑战,但可以提供对一系列应用的有用洞察。为此,我们在本文件中为大型结构结构的对称随机矩阵引入一个无源成份理论总体框架,其中含有各种差异,为结构化的对称随机矩阵(ATE)引入了无源成份理论总框架,为一般线性组合设定了新颖的无源成份扩展,并进一步表明,在一般维格矩阵噪音的假设中,平均矩阵假设结构为低级结构,但可以提供对一系列应用的有用洞察。在某些轻微常规正常条件下,我们为顶峰化成份成份成份群提供了无源扩展的亚源扩展,并表明它们在某些正常化之后是无源正常的。对于总体线性组合,我们为一般成份矩阵组合设置了新型的无源扩展扩展扩展,在此之后,重量矢量矢量媒介可以任意。我们还提供了一种更笼统的泛度理论扩展性理论,用于尖化的统计结构结构模型,作为我们使用的许多模型的模型的模型。