提出新型谱分析理论，解决带依赖随机矩阵的特征向量渐近性质问题。
Asymptotic Theory of Eigenvectors for Latent Embeddings with Generalized Laplacian Matrices
- 构建广义拉普拉斯矩阵框架，统一处理图结构与随机邻接矩阵
- 建立特征向量与特征值的渐近正态性，支持精确推断与不确定性量化
- 适用于高维图数据、网络嵌入等需要统计推断的场景
拉普拉斯矩阵广泛用于刻画图与流形等潜在结构信息，其归一化项引入了依赖性随机矩阵，成为新随机矩阵理论发展的主要瓶颈。本文首次正式定义一类广义（及正则化）拉普拉斯矩阵，包含经典拉普拉斯矩阵和随机邻接矩阵作为特例，并提出针对潜在嵌入的特征向量渐近理论（ATE-GL）。该理论基于广义二次向量方程处理依赖性，结合局部律的高阶渐近展开，建立了显著特征向量与特征值的渐近正态性，可实现对广义拉普拉斯矩阵相关应用的精确推断与不确定性量化。文中讨论了该框架的应用并以数值实验验证其有效性。
原文摘要 · Abstract (English)
Laplacian matrices are commonly employed in many real applications, encoding the underlying latent structural information such as graphs and manifolds. The use of the normalization terms naturally gives rise to random matrices with dependency. It is well-known that dependency is a major bottleneck of new random matrix theory (RMT) developments. To this end, in this paper, we formally introduce a class of generalized (and regularized) Laplacian matrices, which contains the Laplacian matrix and the random adjacency matrix as a specific case, and suggest the new framework of the asymptotic theory of eigenvectors for latent embeddings with generalized Laplacian matrices (ATE-GL). Our new theory is empowered by the tool of generalized quadratic vector equation for dealing with RMT under dependency, and delicate high-order asymptotic expansions of the empirical spiked eigenvectors and eigenvalues based on local laws. The asymptotic normalities established for both spiked eigenvectors and eigenvalues will enable us to conduct precise inference and uncertainty quantification for applications involving the generalized Laplacian matrices with flexibility. We discuss some applications of the suggested ATE-GL framework and showcase its validity through some numerical examples.
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