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Nuclear norm minimization (NNM) has recently gained significant attention forits use in rank minimization problems. Similar to compressed sensing, usingnull space characterizations, recovery thresholds for NNM have been studied in\cite{arxiv,Recht_Xu_Hassibi}. However simulations show that the thresholds arefar from optimal, especially in the low rank region. In this paper we apply therecent analysis of Stojnic for compressed sensing \cite{mihailo} to the nullspace conditions of NNM. The resulting thresholds are significantly better andin particular our weak threshold appears to match with simulation results.Further our curves suggest for any rank growing linearly with matrix size $n$we need only three times of oversampling (the model complexity) for weakrecovery. Similar to \cite{arxiv} we analyze the conditions for weak, sectionaland strong thresholds. Additionally a separate analysis is given for specialcase of positive semidefinite matrices. We conclude by discussing simulationresults and future research directions.

New Null Space Results and Recovery Thresholds for Matrix Rank Minimization