Fast monte-carlo algorithms for finding low-rank approximations

We consider the problem of approximating a given m × n matrix A by another matrix of specified rank k , which is smaller than m and n . The Singular Value Decomposition (SVD) can be used to find the "best" such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this article, we develop an algorithm that is qualitatively faster, provided we may sample the entries of the matrix in accordance with a natural probability distribution. In many applications, such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D * of rank at most k so that holds with probability at least 1 − δ (where |·| F is the Frobenius norm). The algorithm takes time polynomial in k ,1/ϵ, log(1/δ) only and is independent of m and n . In particular, this implies that in constant time, it can be determined if a given matrix of arbitrary size has a good low-rank approximation.