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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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TL;DR: In this paper, the authors define geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings and expose all the parameters involved in the definitions and characterizations and establish relations between them.
Abstract: This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups.

9 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigated the role of metric subregularity in the convergence of Picard iterations of nonexpansive maps in Hilbert spaces and showed that the existence of an error bound is sufficient for strong convergence.
Abstract: We investigate the role of error bounds, or metric subregularity, in the convergence of Picard iterations of nonexpansive maps in Hilbert spaces. Our main results show, on one hand, that the existence of an error bound is sufficient for strong convergence and, on the other hand, that an error bound exists on bounded sets for nonexpansive mappings possessing a fixed point whenever the space is finite dimensional. In the Hilbert space setting, we show that a monotonicity property of the distances of the Picard iterations is all that is needed to guarantee the existence of an error bound. The same monotonicity assumption turns out also to guarantee that the distance of Picard iterates to the fixed point set converges to zero. Our results provide a quantitative characterization of strong convergence as well as new criteria for when strong, as opposed to just weak, convergence holds.

8 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extend the developments in Kruger et al. and characterize the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations.
Abstract: Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples.

8 citations

Posted Content
17 Feb 2019
TL;DR: In this paper, the authors studied general nonlinear transversality properties of collections of sets in normed vector or Banach spaces near a point in their intersection and discussed quantitative relationships between these properties and the corresponding nonlinear regularity properties of set-valued mappings.
Abstract: This paper continues the study of 'good arrangements' of collections of sets in normed vector or Banach spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties, namely, $\varphi-$semitransversality, $\varphi-$subtransversality and $\varphi-$transversality. We focus on primal space (metric and slope) characterizations of these properties. Some characterizations presented in the paper are new also in the Holder (and even linear) setting. We also discuss quantitative relationships between the nonlinear transversality properties of collections of sets and the corresponding nonlinear regularity properties of set-valued mappings as well as nonlinear extensions of the new 'transversality properties of a set-valued mapping to a set in the range space' due to Ioffe.

8 citations

Journal ArticleDOI
TL;DR: In this article , the interference nulling capability of reconfigurable intelligent surface (RIS) in a multiuser environment where multiple single-antenna transceivers communicate simultaneously in a shared spectrum is investigated.
Abstract: This paper investigates the interference nulling capability of reconfigurable intelligent surface (RIS) in a multiuser environment where multiple single-antenna transceivers communicate simultaneously in a shared spectrum. From a theoretical perspective, we show that when the channels between the RIS and the transceivers have line-of-sight and the direct paths are blocked, it is possible to adjust the phases of the RIS elements to null out all the interference completely and to achieve the maximum $K$ degrees-of-freedom (DoF) in the overall $K$ -user interference channel, provided that the number of RIS elements exceeds some finite value that depends on $K$ . Algorithmically, for any fixed channel realization we formulate the interference nulling problem as a feasibility problem, and propose an alternating projection algorithm to efficiently solve the resulting nonconvex problem with local convergence guarantee. Numerical results show that the proposed alternating projection algorithm can null all the interference if the number of RIS elements is only slightly larger than a threshold of $2K(K-1)$ . For the practical sum-rate maximization objective, this paper proposes to use the zero-forcing solution obtained from alternating projection as an initial point for subsequent Riemannian conjugate gradient optimization and shows that it has a significant performance advantage over random initializations. For the objective of maximizing the minimum rate, this paper proposes a subgradient projection method which is capable of achieving excellent performance at low complexity.

8 citations

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