Improved Thresholds for Rank Minimization

Nuclear norm minimization (NNM) has recently gained attention for its use in rank minimization problems. In this paper, we define weak, sectional and strong recovery for NNM to succeed at finding the low rank solution. We find tight conditions for these and analyze them for the case where the linear measurement operator consists of i.i.d. Gaussian entries. Finally we calculate the so called weak, sectional and strong thresholds for the success of nuclear norm minimization. To obtain our results, we generalize the notion of sign and support from sparse vectors to low rank matrices, and achieve a weak threshold which is much closer to the empirical phase transition curve of nuclear norm minimization than the existing bounds available in the literature.