A generalization of wirtinger flow for exact interferometric inversion

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, for which significant advancements were made in recent years despite the ill-posed and nonconvex nature of the problem. One such prominent phase retrieval method is Wirtinger flow (WF) [E. J. Candes, X. Li, and M. Soltanolkotabi, IEEE Trans. Inform. Theory, 61 (2015), pp. 1985-2007], a computationally efficient nonconvex optimization framework that provides high probability guarantees for exact recovery under certain measurement models, specifically 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). Our approach treats the theory of low rank matrix recovery and the nonconvex optimization approach of WF in a unified framework. Such a treatment facilitates the identification of a new sufficient condition on the lifted forward model for exact recovery via GWF and results in a deterministic framework based on geometric arguments for convergence. Thereby, GWF extends the model specific probabilistic guarantees in [12] to arbitrary measurement maps characterized over the equivalent lifted domain in the context of interferometric inversion, covering both random and deterministic measurement models. We then establish our sufficient condition for the cross-correlations of linear measurements collected by complex Gaussian sampling vectors. In the particular case of interferometric inversion with the Gaussian model, we show that the exact recovery theory of standard WF implies our sufficient condition when we have cross-correlations, and the regularity condition of WF is redundant. Finally, we demonstrate the effectiveness of GWF numerically in a deterministic, interferometric multistatic radar imaging scenario.