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.

翻译：计算矩阵的等级是科学计算中的一个基本问题。 互联网生成的数据集往往与高维稀释矩阵的分析相对应。 尽管最近在推广传统单值分解(SVD)方面有所进步,但高维稀释矩阵的等级仍然缺乏有效的估计算法。 在复杂网络的可控性理论的启发下,我们将矩阵的等级转换为最大匹配计算。 然后,我们通过使用洞穴法建立了快速级估测算法,这是计算最大匹配率的有力近似技术,用来估计稀散矩阵的等级。 由于这种方法的自然复杂性较低,我们显示高维度稀释矩阵的等级可以比高精度SVD更快地估计。 我们的方法提供了一种高效的途径,可以快速估计高维稀释矩阵的等级,而SVD计算等级的时间成本是不可接受的。