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Phase retrieval via reweighted Wirtinger flow.

Ziyang Yuan, Hongxia Wang

|April 5, 2017

Summary

Reweighted Wirtinger flow (RWF) improves phase retrieval (PR) by solving sub-problems with adaptive weights. This novel method offers geometric convergence and lower sampling complexity than existing Wirtinger flow (WF) and truncated Wirtinger flow (TWF) techniques.

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Area of Science:

Signal Processing
Computational Imaging
Applied Mathematics

Background:

Phase retrieval (PR) is a critical ill-posed inverse problem with broad applications.
Existing methods like Wirtinger flow (WF) have limitations in convergence and efficiency.
The need for more robust and sample-efficient PR algorithms is evident.

Purpose of the Study:

To introduce a novel Reweighted Wirtinger Flow (RWF) method for solving the phase retrieval problem.
To theoretically and numerically demonstrate the advantages of RWF over existing methods.
To enhance the performance of phase retrieval, particularly in scenarios with limited sampling.

Main Methods:

The proposed Reweighted Wirtinger Flow (RWF) method iteratively solves a series of sub-phase retrieval problems.
Weights are adaptively adjusted throughout the iterative process to guide the search towards the global optimum.
Theoretical analysis establishes geometric convergence properties under specific weight bounds (1 to 10^9).

Main Results:

RWF demonstrates geometric convergence from appropriate initializations.
Numerical simulations confirm that RWF achieves lower sampling complexity compared to the standard WF method.
RWF outperforms the adaptive truncated Wirtinger flow (TWF) method, especially when the ratio of sampling number (m) to signal length (n) is small.

Conclusions:

The Reweighted Wirtinger Flow (RWF) method presents a significant advancement in phase retrieval.
RWF offers improved convergence guarantees and superior sample efficiency.
This method is particularly effective for phase retrieval problems with limited data, outperforming existing adaptive approaches.

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