The minimal measurement number for low-rank matrix recovery

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that 4nr−4r2 generic measurements are both necessary and sufficient for the recovery of rank-r matrices in Cn×n. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound 4nr−4r2 is tight provided n=2k+r,k∈Z+. Motivated by Vinzant's work [19], we construct 11 matrices in R4×4 by computer random search and prove they define injective measurements on rank-1 matrices in R4×4. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2n−1 orthogonal projections are possible for the recovery of x∈Rn from the norm of them, which gives a negative answer for a question raised in [1].