Phase Retrieval via Wirtinger Flow: Theory and Algorithms

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal <inline-formula> <tex-math notation="LaTeX"> \boldsymbol {x}\in \mathbb {C}^{n} </tex-math></inline-formula> about which we have ph...

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Vydáno v:IEEE transactions on information theory Ročník 61; číslo 4; s. 1985 - 2007
Hlavní autoři: Candes, Emmanuel J., Li, Xiaodong, Soltanolkotabi, Mahdi
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.04.2015
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Accuracy
Algorithms
Codes
Coding theory
Computational modeling
Convergence
convergence to global optimum
Diffraction
Fourier transforms
Information retrieval
Information theory
non-convex optimization
Optimization
phase retrieval
Vectors
Wirtinger derivatives
phase retrieval
Non-convex optimization
Wirtinger derivatives
convergence to global optimum
ISSN:0018-9448, 1557-9654
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Shrnutí:We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal <inline-formula> <tex-math notation="LaTeX"> \boldsymbol {x}\in \mathbb {C}^{n} </tex-math></inline-formula> about which we have phaseless samples of the form <inline-formula> <tex-math notation="LaTeX">y_{r} = \left |{\langle \boldsymbol {a}_{r}, \boldsymbol {x} \rangle }\right |^{2} </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">r = 1,\ldots , m </tex-math></inline-formula> (knowledge of the phase of these samples would yield a linear system). This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2015.2399924