We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from an underdetermined system of random linear measurements. Specifically, our results address random Hermitian rank-one measurements as used in a version of the phase retrieval problem; that is, each measurement can be represented as the inner product of the unknown matrix and the outer product of a given realization of the standard complex Gaussian random vector.
We obtain our results by establishing that with high probability the measurement operator consisting of independent realizations of such a random rank-one matrix exhibits the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit ball.
We complement our analysis by numerical experiments comparing the solutions of noise-blind and noise-aware formulations. These experiments confirm that noise-blind optimization methods exhibit comparable robustness to noise-aware formulations.