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Description

We consider the following phase retreival problem: Let $A^*=(\vec{a}_{1},...\vec{a}_{N})'\in\mathbb{C}^{N\times n}$, where $\vec{a}_i$ are sensing vectors, and $b\in\mathbb{R}^{N}$. We want to recover $x$ up to a global phase in $|A^*x|=b; x\in \mathbb{C}^{n}$. It can also be understood as a nonconvex feasibility problem. This problem describes many models such as crystallography and quantum tomography. [11][15] has proven we need at least $N=4n-4$ measurements to ensure the uniqueness of $x$ upto a global phase, without prior knowledge of image (sector constrain). The minimum measurement is reached when $A^*$ has circulation patterns, which will make $A^*$ far from any random matrix.

In this research talk, we will focus on using proposed algorithm (based on Douglas Rachford/ADMM) to solve phase retreival problem in the minimum measurement case. I will give some proven results on fixed point set, local convergence, global convergence, noise stability as well as numerical results.