The Global Landscape of Phase Retrieval Ⅱ:Quotient Intensity Models

摘要

A fundamental problem in phase retrieval is to reconstruct an unknown signal from a set of magnitude-only measurements.In this work we introduce three novel quotient intensity models(QIMs) based on a deep modification of the traditional intensity-based models.A remarkable feature of the new loss functions is that the corresponding geometric landscape is benign under the optimal sampling complexity.When the measurements ai∈Rn are Gaussian random vectors and the number of measurements m≥Cn,the QIMs admit no spurious local minimizers with high probability,i.e.,the target solution x is the unique local minimizer(up to a global phase) and the loss function has a negative directional curvature around each saddle point.Such benign geometric landscape allows the gradient descent methods to find the global solution x(up to a global phase) without spectral initialization.