Towards Sample-Optimal Methods for Solving Random Quadratic Equations with Structure

We consider the problem of estimating a structured high-dimensional parameter vector using random Gaussian quadratic samples. This problem is a generalization of the classical problem of phase retrieval and impacts numerous problems in computational imaging. We provide a generic algorithm based on alternating minimization that, if properly initialized, achieves information-theoretically optimal sample complexity. In essence, we show that solving a system of random quadratic equations with structural constraints is (nearly) as easy as solving the corresponding linear system with the same constraints, if a proper initial guess of the solution is available. As an immediate consequence, our approach improves upon the best known existing sample complexity results for phase retrieval (structured or otherwise). We support our theory via several numerical experiments. A full version of this paper is accessible at: https://gaurijagatap.github.io/assets/ISIT18.pdf.