On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

We consider the problem of recovering a complex vector x ∈ ℂⁿ from m quadratic measurements $\left\{ {\left\langle {{A_i}{\mathbf{x}},{\mathbf{x}}} \right\rangle } \right\}_{i = 1}^m$. This problem, known as quadratic feasibility, encompasses the well known phase retrieval problem and has applications in a wide range of important areas including power system state estimation and x-ray crystallography. In general, not only is the the quadratic feasibility problem NP-hard to solve, but it may in fact be unidentifiable. In this paper, we establish conditions under which this problem becomes identifiable, and further prove isometry properties in the case when the matrices $\left\{ {{A_i}} \right\}_{i = 1}^m$ are Hermitian matrices sampled from a complex Gaussian distribution. Moreover, we explore a nonconvex optimization formulation of this problem, and establish salient features of the associated optimization landscape that enables gradient algorithms with an arbitrary initialization to converge to a globally optimal point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.