Symmetric rank-one updates from partial spectrum with an application to out-of-sample extension

TL;DR

This paper introduces an efficient method for updating a matrix's spectrum when only partial spectral information is available, with applications to extending graph Laplacian eigenvectors to new data points.

Contribution

It proposes a novel scheme for rank-one spectral updates from partial spectrum data and applies it to out-of-sample extension of graph Laplacian eigenvectors.

Findings

The scheme guarantees error bounds for spectral updates.

Application to out-of-sample extension improves eigenvector estimation.

Numerical results demonstrate the method's advantages.

Abstract

Rank-one update of the spectrum of a matrix is a fundamental problem in classical perturbation theory. In this paper, we consider its variant where only part of the spectrum is known. We address this variant using an efficient scheme for updating the known eigenpairs with guaranteed error bounds. Then, we apply our scheme to the extension of the top eigenvectors of the graph Laplacian to a new data sample. In particular, we model this extension as a perturbation problem and show how to solve it using our rank-one updating scheme. We provide a theoretical analysis of this extension method, and back it up with numerical results that illustrate its advantages.