Phase retrieval: uniqueness guarantees, tensorial liftings, and applications

The essential task behind phase retrieval is the recovery of an unknown signal from phaseless data like its Fourier intensity. Although the Fourier transform is a well-understood operator, the loss of its phase turns the signal reconstruction into a severely ill-posed inverse problem, where the strong ambiguousness ranks first of the analytic and algorithmic challenges. In this postdoctoral thesis, we deal with various theoretical and numerical aspects of phase retrieval and phase retrieval-like problems. On the one side, we study phase retrieval problems in specific applications like optical diffraction tomography, super-resolution, or dynamical sampling; on the other side, we consider phase retrieval from the more general perspective of bilinear and quadratic inverse problems. Overall, we derive several uniqueness and recovery guarantees for different phase retrieval instances, establish a novel non-convex regularization theory with corresponding convergence rates, and develop novel numerical techniques, which, for example, are based on tensor-free implementations of singular value thresholding methods.