Fast Monte-Carlo Algorithms for finding low-rank approximations

In several applications, the data consists of an m X n matrix A and it is of interest to find an approximation $\DD$ of a specified rank k to A where, k is much smaller than m and n. Traditional methods like the Singular Value Decomposition (SVD) help us find the ``best'' such approximation. However, these methods take time polynomial in m and n which is often too prohibitive.In this paper, we develop an algorithm which is qualitatively faster, provided we may sample the entries of the matrix according to a natural probability distribution. Indeed, in the applications such sampling is possible.