  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.
