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$ \\boldsymbol {x}\\in \\mathbb {C}^{n}$

about which we have phaseless samples of the form

$y_{r} = \\left |{\\langle \\boldsymbol {a}_{r}, \\boldsymbol {x} \\rangle }\\right |^{2}$

$r = 1,\\ldots , m$

(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. 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Phase Retrieval via Wirtinger Flow: Theory and Algorithms 论文

Phase Retrieval via Wirtinger Flow: Theory and Algorithms · 作者