Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-normminimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature. Further, the recovery error from noisy data is within a constant of three targets: 1) the minimax risk, 2) an ‘oracle’ error that would be available if the column space of the matrix were known, and 3) a more adaptive ‘oracle’ error which would be available with the knowledge of the column space corresponding to the part of the matrix that stands above the noise. Lastly, the error bounds regarding low-rank matrices are extended to provide an error bound when the matrix has full rank with decaying singular values. The analysis in this paper is based on the restricted isometry property (RIP) introduced in [6] for vectors, and in [22] for matrices.