The global landscape of phase retrieval II: quotient intensity models
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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-based models (QIMs) based 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 a_i∈ 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 global 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.
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