Efficient computation of eigenvalues in diffusion maps: A multi-strategy approach

Abstract

In the pursuit of accelerating the computation of k-largest eigenvalues and eigenvectors, this work presents novel methodologies and insights across three main areas. 1) Speed Arnoldi Iteration Up: By observing existing algorithms, we propose an innovative approach that leverages matrix decomposition to accelerate the computation process. The implementation focuses on iteratively computing orthogonal projections and efficiently storing computed vectors in Krylov subspace. 2) Special Case: Eigenvector for =1: We examine a specific scenario concerning Markov matrices, where the largest eigenvalue is 1. The work provides a detailed proof and analysis of this property, contributing to a deeper understanding of eigenvalues and eigenvectors in the context of stochastic processes. 3) Gaussian Approximation for Markov Matrices: This section delves into the Gaussian approximation for Markov matrices, denoted by P. The work covers theoretical insights, practical challenges, computational efficiency, and empirical validation, providing a comprehensive exploration of this critical method. Together, these sections form a cohesive study aimed at enhancing the computational efficiency of significant algorithms within the field of dimensionality reduction and matrix analysis. The findings may find broad applications in various domains, including image segmentation, speaker verification, anomaly detection, and more.