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Journal ArticleDOI

On Local Convergence of the Method of Alternating Projections

01 Apr 2016-Foundations of Computational Mathematics (Springer US)-Vol. 16, Iss: 2, pp 425-455
TL;DR: In this article, the authors proved local convergence of alternating projections between subanalytic sets under a mild regularity hypothesis on one of the sets, and showed that the speed of convergence is O(k √ √ σ(k − σ ) for some constant σ √ n, σ (n) for some σ σ = (0, √ N) √ (n − ρ) for any σ > 0.
Abstract: The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets $$A,B$$A,B under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is $${\mathcal {O}}(k^{-\rho })$$O(k-?) for some $$\rho \in (0,\infty )$$??(0,?).

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Citations
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Journal ArticleDOI
TL;DR: In this article, the authors investigated the regularity properties of set-valued mappings in variational analysis and their applications in convergence analysis and optimisation, and showed that these properties are useful for stability analysis of optimisation and variational problems.
Abstract: Regularity properties lie at the core of variational analysis because of their importance for stability analysis of optimisation and variational problems, constraint qualifications, qualification conditions in coderivative and subdifferential calculus and convergence analysis of numerical algorithms [3–5, 7, 21, 23]. The thesis is devoted to investigation of several research questions related to regularity properties in variational analysis and their applications in convergence analysis and optimisation. Following the works by Kruger [13, 14], we examine several useful regularity properties of collections of sets in both linear and Hölder-type settings and establish their characterisations and relationships to regularity properties of set-valued mappings [16, 17]. Following the recent publications by Lewis et al. [19, 20], Drusvyatskiy et al. [6] and some others, we study application of the uniform regularity and related properties of collections of sets to alternating projections for solving nonconvex feasibility problems and compare existing results on this topic [15, 18, 22]. Motivated by Ioffe [8] and his subsequent publications [9–11], we use the classical iteration scheme going back to Banach, Schauder, Lyusternik and Graves to establish criteria for regularity properties of set-valued mappings and compare this approach with the one based on the Ekeland variational principle [12]. Finally, following the recent works by Anh and Khanh [1] on stability analysis for optimisation-related problems, we investigate calmness of set-valued solution mappings of variational problems [2].
Journal ArticleDOI
TL;DR: In this article, the convergence of alternating projections between non-convex sets is studied and applications to convergence of the Gerchberg-Saxton error reduction method, of the Gaussian expectation-maximization algorithm, and of Cadzow's algorithm are discussed.
Abstract: We consider convergence of alternating projections between non-convex sets and obtain applications to convergence of the Gerchberg-Saxton error reduction method, of the Gaussian expectation-maximization algorithm, and of Cadzow’s algorithm.
Proceedings ArticleDOI
01 Jun 2020
TL;DR: In this paper, an OAM mode detection method based on Gerchberg-Saxton (GS) algorithm was proposed for free-space optical (FSO) communications, which is able to restore the vortex phase of multi-mode multiplexed Laguerre-Gaussian (LG) beams in weak and medium atmospheric turbulence cases.
Abstract: In free-space optical (FSO) communications, orbital angular momentum (OAM) provides extra degrees of freedom to improve spectrum efficiency by multiplexing a set of orthogonal modes on the same wavelength channel. To recover the information carried on the optical OAM modes correctly, the receiver has to first know which OAM modes are transmitted. In this paper, we first present the atmospheric turbulence channel model and the FSO system model. Then, we propose an OAM mode detection method based on Gerchberg-Saxton (GS) algorithm from the perspective of optical signal processing, rather than using additional optical components. Simulation results show that the proposed GS algorithm-based detection method is able to restore the vortex phase of multi-mode multiplexed Laguerre-Gaussian (LG) beams in weak and medium atmospheric turbulence cases, and the optical OAM modes could be detected with the recovered vortex phase.
Book ChapterDOI
01 Jan 2018
TL;DR: This chapter is devoted to the exposition and developments of the fundamental principles of variational analysis, which play a crucial role in resolving many issues in variational theory and applications by employing optimization ideas and techniques.
Abstract: This chapter is devoted to the exposition and developments of the fundamental principles of variational analysis, which play a crucial role in resolving many issues of variational theory and applications by employing optimization ideas and techniques.
Posted Content
TL;DR: In this article, the authors extend the previous convergence results on the generalized alternating projection method, from subspaces, to include smooth manifolds, and show that locally it will behave in the same way, with the same rate as predicted in [arXiv:1703.10547].
Abstract: In this paper we extend the previous convergence results on the generalized alternating projection method, from subspaces [arXiv:1703.10547], to include smooth manifolds. We show that locally it will behave in the same way, with the same rate as predicted in [arXiv:1703.10547]. The goal is to get closer to a rate for general convex sets, where convergence, but not rate is known. If a finite identification property can be shown for two convex sets, to locally smooth manifolds, then the rates from this paper also apply to those sets. We present a few examples where this is the case, and also a counter example for when this is not the case.
References
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Journal ArticleDOI
TL;DR: Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods and it is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm forThe problem of two intensity measurements converge.
Abstract: Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

5,210 citations

Journal Article
01 Jan 1972-Optik
TL;DR: In this article, an algorithm is presented for the rapid solution of the phase of the complete wave function whose intensity in the diffraction and imaging planes of an imaging system are known.

5,197 citations

Journal ArticleDOI
TL;DR: This work studies two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators with real-time requirements.
Abstract: Splitting algorithms for the sum of two monotone operators.We study two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators. These algorithms ar...

1,939 citations

Journal ArticleDOI
22 Jul 1999-Nature
TL;DR: Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens was proposed in this paper, where the authors extended the methodology to allow the imaging of micro-scale specimens.
Abstract: Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens

1,791 citations

Journal ArticleDOI
TL;DR: A very broad and flexible framework is investigated which allows a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence in convex feasibility problems.
Abstract: Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.

1,742 citations