Updating Singular Value Decomposition for Rank One Matrix Perturbation

Abstract

An efficient Singular Value Decomposition (SVD) algorithm is an important tool for distributed and streaming computation in big data problems. It is observed that update of singular vectors of a rank-1 perturbed matrix is similar to a Cauchy matrix-vector product. With this observation, in this paper, we present an efficient method for updating Singular Value Decomposition of rank-1 perturbed matrix in $O(n^2 \ \text{log}(\frac{1}ε))$ time. The method uses Fast Multipole Method (FMM) for updating singular vectors in $O(n \ \text{log} (\frac{1}ε))$ time, where $ε$ is the precision of computation.