The minimal measurement number for low-rank matrices recovery
This work resolves a fundamental question in compressed sensing and matrix recovery, providing tight bounds that guide practical measurement design for low-rank matrix recovery.
The paper determines the minimal number of measurements needed for low-rank matrix recovery, proving that 4nr-4r^2 generic measurements are necessary and sufficient for complex matrices, and showing the real case requires fewer measurements, disproving a previous conjecture. It also applies these results to phase retrieval, showing fewer than 2n-1 orthogonal projections suffice.
The paper presents several results that address a fundamental question in low-rank matrices recovery: how many measurements are needed to recover low rank matrices? We begin by investigating the complex matrices case and show that $4nr-4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-$r$ matrices in $\C^{n\times n}$ by algebraic tools. 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 $4nr-4r^2$ is tight provided $n=2^k+r, k\in \Z_+$. Motivated by Vinzant's work, we construct $11$ matrices in $\R^{4\times 4}$ by computer random search and prove they define injective measurements on rank-$1$ matrices in $\R^{4\times 4}$. 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 $x\in \R^n$ from the norm of them.