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

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.