A Generalization of Wirtinger Flow for Exact Interferometric Inversion

Abstract

Interferometric inversion involves recovery of a signal from cross-correlations of its linear transformations. A close relative of interferometric inversion is the generalized phase retrieval problem, which consists of recovering a signal from the auto-correlations of its linear transformations. Recently, significant advancements have been made in phase retrieval methods despite the ill-posed, and non-convex nature of the problem. One such method is Wirtinger Flow (WF), a non-convex optimization framework that provides high probability guarantees of exact recovery under certain measurement models, such as coded diffraction patterns, and Gaussian sampling vectors. In this paper, we develop a generalization of WF for interferometric inversion, which we refer to as Generalized Wirtinger Flow (GWF). GWF theory extends the probabilistic exact recovery results of WF to arbitrary measurement models characterized in the equivalent lifted problem, hence covers a larger class of measurement models. Our framework unifies the theory of low rank matrix recovery (LRMR) and the non-convex optimization approach of WF, thereby establishes theoretical advantages of the non-convex approach over LRMR. We identify a new sufficient condition on the lifted forward model that directly implies exact recovery conditions of standard WF. This condition is less stringent than those of LRMR, which is the state of the art approach for exact interferometric inversion. We establish our sufficient condition for the cross-correlations of linear measurements collected by complex Gaussian sampling vectors, and show that the exact recovery conditions of standard WF imply our sufficient condition. As a result, the regularity condition of WF becomes redundant in solving the interferometric inversion problem. Finally, we demonstrate the effectiveness of GWF numerically in a deterministic multi-static radar imaging scenario.