Phase retrieval is the recovery of a signal from only the magnitudes, and not the phases, of complex-valued linear measurements. Phase retrieval problems arise in many different applications, particularly in crystallography and microscopy. Mathematically, phase retrieval recovers a complex valued signal \(x\in \mathbb{C}^n\) from \(m\) measurements of the form
Phase retrieval deals with the recovery of an \(n\)-dimensional signal \(x^0\in\mathbb{C}^n\), from \(m\geq n\) magnitude measurements of the form \begin{align} \label{eq:original} b_i = |\langle a_i, x^0\rangle|, \quad i = 1,2,\ldots,m, \end{align} where \(a_i\in\mathbb{C}^n\), and \(i=1,2,\ldots,m\) are measurement vectors. While the recovery of \(x^0\) from these non-linear measurements is non-convex, it can be convexified via “lifting” methods that convert the phase retrieval problem to a semidefinite program (SDP). But convexity comes at a high cost: lifting methods square the dimensionality of the problem.
