Reduction of the Rank Calculation of a High-Dimensional Sparse Matrix Based on Network Controllability Theory

Abstract

Numerical computing of the rank of a matrix is a fundamental problem in scientific computation. The datasets generated by the Internet often correspond to the analysis of high-dimensional sparse matrices. Notwithstanding recent advances in the promotion of traditional singular value decomposition (SVD), an efficient estimation algorithm for the rank of a high-dimensional sparse matrix is still lacking. Inspired by the controllability theory of complex networks, we converted the rank of a matrix into maximum matching computing. Then, we established a fast rank estimation algorithm by using the cavity method, a powerful approximate technique for computing the maximum matching, to estimate the rank of a sparse matrix. In the merit of the natural low complexity of the cavity method, we showed that the rank of a high-dimensional sparse matrix can be estimated in a much faster way than SVD with high accuracy. Our method offers an efficient pathway to quickly estimate the rank of the high-dimensional sparse matrix when the time cost of computing the rank by SVD is unacceptable.