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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 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 $ a_i\in \mathbb{R}^n$ are Gaussian random vectors and the number of measurements $m\ge Cn$, 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.
}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2021-0010}, url = {http://global-sci.org/intro/article_detail/aam/20173.html} }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 $ a_i\in \mathbb{R}^n$ are Gaussian random vectors and the number of measurements $m\ge Cn$, 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.