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| The Global Landscape of Phase Retrieval II: Quotient Intensity Models |
| Jian-Feng Cai,Meng Huang,Dong Li,Yang Wang |
| (Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong;SUSTech International Center for Mathematics and Department of Mathematics, Southern University of Science and Technology, Shenzhen,
Guangdong 518055, China) |
| DOI： |
| Abstract: |
| A fundamental problem in phase retrieval is to reconstruct an un known signal from a set of magnitude-only measurements. In this work we
introduce three novel quotient intensity models (QIMs) based on a deep modi fication 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 ∈ R
n are Gaus sian 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. |
| Key words: Phase retrieval, landscape analysis, non-convex optimization. |