Matrix Recovery Via Symmetric Rank-one Measurements With Random Unit-modulus Vectors

Unit-modulus measurement vectors, due to their phase-only variations, exhibit favorable hardware compatibility and storage efficiency. In this paper, we consider matrix recovery using symmetric unit-modulus rank-one measurements. We identify a fundamental limitation of symmetric unit-modulus measurements—their inability to capture individual diagonal entries, and show that matrix recovery can be achieved using such measurements when the diagonal entries are known or directly measurable. We construct a stacked operator and establish that it satisfies a mixed-norm restricted isometry property ($\mathrm{mRIP}$) for unit-modulus measurements. Leveraging the $\mathrm{mRIP}$ condition, we derive exact and stable recovery results for matrix recovery with unit-modulus measurements in both noiseless and noisy settings. Our work is the first to provide the theoretical foundation for matrix recovery under symmetric rank-one unit-modulus measurements. Numerical experiments further corroborate these theoretical findings.