Non-Convex Structured Phase Retrieval
This addresses the challenge of sample-efficient signal recovery in domains like optics and imaging, but it is incremental as it builds on existing structural assumptions.
The paper tackles the phase retrieval problem by leveraging structural assumptions like sparsity or low-rank models to reduce measurement requirements, focusing on non-convex approaches with sample complexity guarantees.
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, sub-diffraction 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 over-complete 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 (i) sparsity of a given signal/image or (ii) a low rank 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 non-convex 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.