TY - JOUR
T1 - The Global Landscape of Phase Retrieval II: Quotient Intensity Models
AU - Cai , Jian-Feng
AU - Huang , Meng
AU - Li , Dong
AU - Wang , Yang
JO - Annals of Applied Mathematics
VL - 1
SP - 62
EP - 114
PY - 2022
DA - 2022/01
SN - 38
DO - http://doi.org/10.4208/aam.OA-2021-0010
UR - https://global-sci.org/intro/article_detail/aam/20173.html
KW - Phase retrieval, landscape analysis, non-convex optimization.
AB - <p style="text-align: justify;">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&nbsp; the new loss functions is that the corresponding geometric landscape is benign under the optimal sampling complexity.&nbsp; 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.</p>