Fast Rank-One Alternating Minimization Algorithm for Phase Retrieval

The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector x ~ ∈ C d from a set of N measurements b n = | f n ∗ x ~ | 2 , n = 1 , … , N , where { f n } n = 1 N forms a frame of C d . Existing algorithms usually use a...

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Vydáno v:Journal of scientific computing Ročník 79; číslo 1; s. 128 - 147
Hlavní autoři: Cai, Jian-Feng, Liu, Haixia, Wang, Yang
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.04.2019
Springer Nature B.V
Témata:
Algorithms
Computational Mathematics and Numerical Analysis
Convergence
Convex analysis
Efficiency
Fourier transforms
Hilbert space
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Optimization
Phase retrieval
Polynomials
Splitting
Theoretical
Alternating minimization
Non-convex optimizaton
Alternating gradient descent
Phase retrieval
Rank-one
ISSN:0885-7474, 1573-7691
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Shrnutí:The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector x ~ ∈ C d from a set of N measurements b n = | f n ∗ x ~ | 2 , n = 1 , … , N , where { f n } n = 1 N forms a frame of 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.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-018-0857-9