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Matrices with low-rank structure are ubiquitous in scientific computing.Choosing an appropriate rank is a key step in many computational algorithmsthat exploit low-rank structure. However, estimating the rank has been donelargely in an ad-hoc fashion in large-scale settings. In this work we develop arandomized algorithm for estimating the numerical rank of a (numericallylow-rank) matrix. The algorithm is based on sketching the matrix with randommatrices 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.We prove a result on the accuracy of the sketched singular values and show thatgaps in the spectrum are detected. For an $m\times n$ $(m\geq n)$ matrix ofnumerical rank $r$, the algorithm runs with complexity $O(mn\log n+r^3)$, orless for structured matrices. The steps in the algorithm are required as a partof many low-rank algorithms, so the additional work required to estimate therank can be even smaller in practice. Numerical experiments illustrate thespeed and robustness of our rank estimator.

Fast randomized numerical rank estimation for numerically low-rank matrices