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.

翻译：拉普拉斯矩阵广泛应用于诸多实际应用中，用于编码图与流形等潜在结构信息。归一化项的使用自然引入了具有依赖性的随机矩阵。众所周知，依赖性是随机矩阵理论新发展的主要瓶颈。为此，本文正式引入一类广义（且正则化的）拉普拉斯矩阵——其特例包含拉普拉斯矩阵与随机邻接矩阵，并提出基于广义拉普拉斯矩阵的潜在嵌入特征向量渐近理论新框架。该理论通过广义二次向量方程工具处理依赖条件下的随机矩阵问题，并基于局部定律对经验尖峰特征向量与特征值进行精细的高阶渐近展开。所建立的尖峰特征向量与特征值的渐近正态性，将使我们能够灵活地对涉及广义拉普拉斯矩阵的应用进行精确推断与不确定性量化。本文讨论了所提理论框架的若干应用，并通过数值算例验证了其有效性。