Fast estimation method for rank of a high-dimensional sparse matrix

Numerical computing the rank of a matrix is a fundamental problem in scientific computation. The data sets generated by Internet often correspond to the analysis of high-dimensional sparse matrices. Notwithstanding the recent advances in the promotion of traditional singular value decomposition (SVD), an efficient estimation algorithm for rank of a high-dimensional sparse matrix is still lacked. Inspired by the controllability theory of complex networks, we converted the rank of a matrix into max-matching computing. Then we established a fast rank estimation algorithm by using cavity method, a powerful approximate technique for computing the max-matching, to estimate the rank of a sparse matrix. In the merit of its natural low complexity of 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 fast estimate the rank of the high-dimensional sparse matrix, when the time cost of computing the rank by SVD is unacceptable.