Iteration on single vector for extracting two extremal eigenpairs of symmetric matrices

Abstract

It has been shown that the first two extremal eigenpairs of symmetric matrices are involved a lot in spectral clustering, dimensionality reduction, image segmentation, and graph theory. We also know that the eigengap directly affects the stability of principal eigenvector related algorithms, such as the Hyperlink-Induced Topic Search (HITS). To extract two extremal eigenpairs, conventional methods are generally iterated on two orthogonal vectors, i.e., 2-dimensional subspace. This paper introduces an approach that is iterated on single vector but can simultaneously extract the largest or smallest two eigenpairs of general symmetric matrices, which reduces the solution scale by half. The complete stability analysis is also presented here. Numerical experiments demonstrate the superior performance of the proposed algorithm.