Improved testing of low rank matrices

We study the problem of determining if an input matrix A εR^{m x n} can be well-approximated by a low rank matrix. Specifically, we study the problem of quickly estimating the rank or stable rank of A, the latter often providing a more robust measure of the rank. Since we seek significantly sublinear time algorithms, we cast these problems in the property testing framework. In this framework, A either has low rank or stable rank, or is far from having this property. The algorithm should read only a small number of entries or rows of A and decide which case A is in with high probability. If neither case occurs, the output is allowed to be arbitrary. We consider two notions of being far: (1) A requires changing at least an ε-fraction of its entries, or (2) A requires changing at least an ε-fraction of its rows. We call the former the "entry model" and the latter the "row model". We show:We also give an empirical evaluation of our rank and stable rank algorithms on real and synthetic datasets.