Asymptotic Theory of Eigenvectors for Latent Embeddings With Generalized Laplacian Matrices

Laplacian matrices are widely used in practice to capture latent structural information in data, ranging from graphs to manifolds. Their normalization forms naturally induce dependencies among matrix entries, and such dependencies are known to pose significant challenges for advances in random matrix theory (RMT). Motivated by this, we introduce a general class of generalized Laplacian matrices, which includes both the standard Laplacian and random adjacency matrices as special cases, and we develop a new framework—Asymptotic Theory of Eigenvectors for latent embeddings with Generalized Laplacian matrices (ATE-GL)—for studying their spectral properties. Our theory is driven by two key ingredients: the use of generalized quadratic vector equations to handle dependency in RMT, and refined high-order asymptotic expansions for empirical spiked eigenvectors and eigenvalues based on local laws. These results lead to asymptotic normality for both spiked eigenvectors and eigenvalues, enabling precise statistical inference and uncertainty quantification for a broad class of applications involving generalized Laplacian matrices. We also discuss two motivating applications of the ATE-GL framework and demonstrate its effectiveness through numerical examples.