Fast randomized numerical rank estimation

Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an ad-hoc fashion in previous studies. In this work we develop a randomized algorithm for estimating the numerical rank of a matrix. The algorithm is based on sketching the matrix with random matrices from both left and right; the key fact is that with high probability, the sketches preserve the orders of magnitude of the leading singular values. The rank can hence be taken to be the number of singular values of the sketch that are larger than the prescribed threshold. For an $m\times n$ $(m\geq n)$ matrix of numerical rank $r$, the algorithm runs with complexity $O(mn\log n+r^3)$, or less for structured matrices. The steps in the algorithm are required as a part of many low-rank algorithms, so the additional work required to estimate the rank can be even smaller in practice. Numerical experiments illustrate the speed and robustness of our rank estimator.