On Local Convergence of the Method of Alternating Projections
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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TL;DR: In this article, the authors show that computationally-cheap inexact projections may suffice instead of exact projection onto nonconvex sets, if one set is defined by sufficiently regular smooth constraints, then projecting onto the approximation obtained by linearizing those constraints around the current iterate suffices.
Abstract: Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically intractable, but we show that computationally-cheap inexact projections may suffice instead. In particular, if one set is defined by sufficiently regular smooth constraints, then projecting onto the approximation obtained by linearizing those constraints around the current iterate suffices. On the other hand, if one set is a smooth manifold represented through local coordinates, then the approximate projection resulting from linearizing the coordinate system around the preceding iterate on the manifold also suffices.
3 citations
01 Nov 2018
TL;DR: This paper investigates the pliable index coding problem, and proposes a novel sparse and low-rank optimization framework to assist efficient algorithms design in real field, thereby minimizing the number of channel uses for message delivery.
Abstract: In this paper, we investigate the pliable index coding problem, where clients are interested in receiving any messages (instead of specific messages) that they do not have. The motivating applications including caching networks, recommendation systems and distributed computing systems, where the clients are happy to receive any messages not available in them. However, the pliable index coding problem turns out to be computationally intractable, for which we propose a novel sparse and low-rank optimization framework to assist efficient algorithms design in real field, thereby minimizing the number of channel uses for message delivery. To address the nonconvex challenges in this framework, we further propose the alternating projection algorithm to solve the sparse and low-rank optimization problem with local convergence guarantees. Simulation results demonstrate that the number of channel uses can be significantly reduced for message delivery via the sparse and low-rank optimization.
3 citations
TL;DR: In this paper, a method to obtain a tight frame by appending some vectors with prescribed norms is presented, based on the alternating projection technique, which is an iterative process.
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TL;DR: In this paper, the core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analyzed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining 'extremal' statements.
Abstract: The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analyzed and refined, leading to a unifying theory, encompassing all existing approaches to obtaining 'extremal' statements. For that, we examine and clarify quantitative relationships between the parameters involved in the respective definitions and statements. Some new characterizations of extremality properties are obtained.
3 citations
23 May 2021
TL;DR: A novel approach to wavefront sensing that uses a non-planar reference that is designed to be brighter and significantly more robust to noise compared to PS-PDI, and is validated using a suite of simulated and real results.
Abstract: One of the classical results in wavefront sensing is phase-shifting point diffraction interferometry (PS-PDI), where the phase of a wavefront is measured by interfering it with a planar reference created from the incident wave itself. The limiting drawback of this approach is that the planar reference, often created by passing light through a narrow pinhole, is dim and noise sensitive. We address this limitation with a novel approach called ReWave that uses a non-planar reference that is designed to be brighter. The reference wave is designed in a specific way that would still allow for analytic phase recovery, exploiting ideas of sparse phase retrieval algorithms. ReWave requires only four image intensity measurements and is significantly more robust to noise compared to PS-PDI. We validate the robustness and applicability of our approach using a suite of simulated and real results.
3 citations
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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•
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
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
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
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