A Spectral Estimation Framework for Phase Retrieval via Bregman Divergence Minimization

Abstract: In this paper, we develop a novel framework to optimally design spectral estimators for phase retrieval given measurements realized from an arbitrary model. We begin by deconstructing spectral methods, and identify the fundamental mechanisms that inherently promote the accuracy of estimates. We then propose a general formalism for spectral estimation as approximate Bregman loss minimization in the range of the lifted forward model that is tractable by a search over rank-1, PSD matrices. Essentially, by the Bregman loss approach we transcend the Euclidean sense alignment based similarity measure between phaseless measurements in favor of appropriate divergence metrics over $\mathbb{R}^M_+$. To this end, we derive spectral methods that perform approximate minimization of KL-divergence, and the Itakura-Saito distance over phaseless measurements by using element-wise sample processing functions. As a result, our formulation relates and extends existing results on model dependent design of optimal sample processing functions in the literature to a model independent sense of optimality. Numerical simulations confirm the effectiveness of our approach in problem settings under synthetic and real data sets.