Solving Complex Quadratic Systems with Full-Rank Random Matrices

We tackle the problem of recovering a complex signal ${\boldsymbol{x}}\in \mathbb{C}^n$ from quadratic measurements of the form $y_i={\boldsymbol{x}}^*{\boldsymbol{A}}_i{\boldsymbol{x}}$ , where ${\boldsymbol{A}}_i$ is a full-rank, complex random measurement matrix whose entries are generated from a rotation-invariant sub-Gaussian distribution. We formulate it as the minimization of a nonconvex loss. This problem is related to the well understood phase retrieval problem where the measurement matrix is a rank-1 positive semidefinite matrix. Here we study the general full-rank case which models a number of key applications such as molecular geometry recovery from distance distributions and compound measurements in phaseless diffractive imaging. Most prior works either address the rank-1 case or focus on real measurements. The several papers that address the full-rank complex case adopt the computationally-demanding semidefinite relaxation approach. In this paper we prove that the general class of problems with rotation-invariant sub-Gaussian measurement models can be efficiently solved with high probability via the standard framework comprising a spectral initialization followed by iterative Wirtinger flow updates on a nonconvex loss. Numerical experiments on simulated data corroborate our theoretical analysis.