Nonconvex Structured Phase Retrieval: A Focus on Provably Correct Approaches

Phase retrieval (PR), also sometimes referred to as quadratic sensing, is a problem that occurs in numerous signal and image acquisition domains ranging from optics, X-ray crystallography, Fourier ptychography, subdiffraction imaging, and astronomy. In each of these domains, the physics of the acquisition system dictates that only the magnitude (intensity) of certain linear projections of the signal or image can be measured. Without any assumptions on the unknown signal, accurate recovery necessarily requires an overcomplete set of measurements. The only way to reduce the measurements/sample complexity is to place extra assumptions on the unknown signal/image. A simple and practically valid set of assumptions is obtained by exploiting the structure inherently present in many natural signals or sequences of signals.

Phase retrieval (PR), also sometimes referred to as quadratic sensing, is a problem that occurs in numerous signal and image acquisition domains ranging from optics, X-ray crystallography, Fourier ptychography, subdiffraction imaging, and astronomy. In each of these domains, the physics of the acquisition system dictates that only the magnitude (intensity) of certain linear projections of the signal or image can be measured. Without any assumptions on the unknown signal, accurate recovery necessarily requires an overcomplete set of measurements. The only way to reduce the measurements/sample complexity is to place extra assumptions on the unknown signal/image. A simple and practically valid set of assumptions is obtained by exploiting the structure inherently present in many natural signals or sequences of signals.

Two commonly used structural assumptions are: 1) the sparsity of a given signal/image or 2) a low-rank (LR) model on the matrix formed by a set, e.g., a time sequence, of signals/images. Both have been explored for solving the PR problem in a sample-efficient fashion. This article describes this work, with a focus on nonconvex approaches that come with sample complexity guarantees under simple assumptions. We also briefly describe other different types of structural assumptions that have been used in recent literature.