Geometric Analysis of Phase Retrieval

Explore the geometric analysis of phase retrieval in this 52-minute lecture by John Wright at ICBS2024. Delve into the generalized phase retrieval (GPR) problem, examining whether it's possible to recover a complex signal from its Fourier magnitudes or reconstruct a length-n complex vector from a set of m measurements. Investigate the effectiveness of nonconvex heuristics in practical GPR applications and the theoretical explanations behind their success. Learn about a least-squares formulation for GPR that exhibits a benign geometric structure under certain conditions, including the absence of spurious local minimizers and negative curvature around saddle points. Discover how this structure enables efficient global minimization using iterative optimization methods without special initialization. Gain insights into the connections between GPR and other optimization problems with similar geometric properties, such as dictionary learning and deconvolution.