Asymptotic Theory of Eigenvectors for Latent Embeddings with Generalized Laplacian Matrices

2025-03-01

AI Summary

This paper addresses the lack of asymptotic theory for latent embedding eigenvectors of generalized Laplacian matrices—particularly when nodes exhibit dependence, rendering classical random matrix theory (RMT) inapplicable. We propose the first high-order asymptotic theory for spiked eigenvectors and eigenvalues under dependence (ATE-GL). Methodologically, we establish the first higher-order eigenvector asymptotic expansion and asymptotic normality for generalized regularized Laplacian matrices, integrating generalized quadratic vector equations, local laws, and dependence-aware RMT tools. Our theory relaxes the stringent independence assumption of classical RMT, enabling precise quantification of embedding uncertainty. Numerical experiments demonstrate high accuracy and robustness of ATE-GL in graph embedding and manifold learning tasks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Graphical Models
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Develops asymptotic theory for eigenvectors in latent embeddings using generalized Laplacian matrices.
• Addresses dependency issues in random matrix theory with new framework ATE-GL.
• Enables precise inference and uncertainty quantification for applications involving generalized Laplacians.

Innovation

• Generalized Laplacian matrices for latent embeddings
• Asymptotic theory using quadratic vector equations
• High-order expansions for eigenvectors and eigenvalues