Journal of Scientific Computing

, Volume 79, Issue 1, pp 128–147 | Cite as

Fast Rank-One Alternating Minimization Algorithm for Phase Retrieval

  • Jian-Feng Cai
  • Haixia LiuEmail author
  • Yang Wang


The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector \({\tilde{{\mathbf {x}}}}\in {\mathbb {C}}^d\) from a set of N measurements \(b_n=|{\mathbf {f}}^*_n{\tilde{{\mathbf {x}}}}|^2,\ n=1,\ldots , N\), where \(\{{\mathbf {f}}_n\}_{n=1}^N\) forms a frame of \({\mathbb {C}}^d\). Existing algorithms usually use a least squares fitting to the measurements, yielding a quartic polynomial minimization. In this paper, we employ a new strategy by splitting the variables, and we solve a bi-variate optimization problem that is quadratic in each of the variables. An alternating gradient descent algorithm is proposed, and its convergence for any initialization is provided. Since a larger step size is allowed due to the smaller Hessian, the alternating gradient descent algorithm converges faster than the gradient descent algorithm (known as the Wirtinger flow algorithm) applied to the quartic objective without splitting the variables. Numerical results illustrate that our proposed algorithm needs less iterations than Wirtinger flow to achieve the same accuracy.


Phase retrieval Alternating minimization Alternating gradient descent Rank-one Non-convex optimizaton 



The authors would like to thank Emmanuel Candès, Mo Mu and Aditya Viswanathan for very helpful discussions.


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Authors and Affiliations

  1. 1.Department of MathematicsThe Hong Kong University of Science and TechnologyKowloonHong Kong
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina

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