On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems
This work addresses a fundamental problem in signal processing with applications in areas like power systems and crystallography, providing theoretical guarantees for recovery and optimization, though it is incremental in building on existing phase retrieval research.
The paper tackles the quadratic feasibility problem, which includes phase retrieval, by establishing identifiability conditions and proving isometry properties for Gaussian-sampled Hermitian matrices, and shows that gradient algorithms can converge to globally optimal points with high probability under certain sample complexity requirements.
We consider the problem of recovering a complex vector $\mathbf{x}\in \mathbb{C}^n$ from $m$ quadratic measurements $\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m$. 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 $\{A_i\}_{i=1}^m$ 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 \emph{globally optimal} point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.