对称随机矩阵的统一$\ell_{2\rightarrow\infty}$特征空间扰动理论
Unified $\ell_{2\rightarrow\infty}$ Eigenspace Perturbation Theory for Symmetric Random Matrices
摘要：
精细矩阵谱扰动理论在现代统计学、计算机科学和网络科学中的应用具有巨大的价值。本文给出了对称随机矩阵的一般$\ell_{2\rightarrow\infty}$特征空间扰动界，它具有独立或相依元素和相当灵活的元素分布。特别是，我们将我们的通用绑定到二进制随机矩阵的独立条目或具有一定的依赖结构，包括非规范化的拉普拉斯算子的非齐次随机图和$m$依赖矩阵。通过详细的比较，我们发现，对于具有独立条目的二元随机矩阵，我们的$\ell_{2\rightarrow\infty}$界比我们所知道的所有现有的界限更紧，而我们的条件是弱于所有，但其中之一，在一个不太常见的制度。我们将我们的扰动界应用于三个问题，并改进了现有技术：稀疏随机图的谱范数的集中，随机块模型中社区的精确恢复和分裂层次聚类的部分一致性。最后，我们讨论了我们的理论对具有更复杂依赖结构和非二元项的随机矩阵、非对称矩形矩阵和其他度量中的诱导扰动理论的推广.
英文摘要：
Modern applications in statistics, computer science and network science have seen tremendous values of finer matrix spectral perturbation theory. In this paper, we derive a generic $\ell_{2\rightarrow\infty}$ eigenspace perturbation bound for symmetric random matrices, with independent or dependent entries and fairly flexible entry distributions. In particular, we apply our generic bound to binary random matrices with independent entries or with certain dependency structures, including the unnormalized Laplacian of inhomogenous random graphs and $m$-dependent matrices. Through a detailed comparison, we found that for binary random matrices with independent entries, our $\ell_{2\rightarrow\infty}$ bound is tighter than all existing bounds that we are aware of, while our condition is weaker than all but one of them in a less common regime. We apply our perturbation bounds in three problems and improve the state of the art: concentration of the spectral norm of sparse random graphs, exact recovery of communities in stochastic block models and partial consistency of divisive hierarchical clustering. Finally we discuss the extensions of our theory to random matrices with more complex dependency structures and non-binary entries, asymmetric rectangular matrices and induced perturbation theory in other metrics.
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