On the minimal number of measurements in low-rank matrix recovery

Abstract

In this paper we present a new way to obtain a bound on the number of measurements sampled from certain distributions that guarantee uniform stable and robust recovery of low-rank matrices. The recovery guarantees are characterized by a stable and robust version of the null space property and verifying this condition can be reduced to the problem of obtaining a lower bound for a quantity of the form inf x∊T ||Ax||2. Gordon's escape through a mesh theorem provides such a bound with explicit constants for Gaussian measurements. Mendelson's small ball method allows to cover the significantly more general case of measurements generated by independent identically distributed random variables with finite fourth moment.