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,?).
Citations
More filters
Posted Content•
TL;DR: In this paper, the convergence of the Douglas-Rachford algorithm with respect to the set of fixed points is characterized in terms of general fixed point operators and local linear convergence results are established for nonconvex feasibility.
Abstract: This paper proposes an algorithm for solving structured optimization problems, which covers both the backward-backward and the Douglas-Rachford algorithms as special cases, and analyzes its convergence. The set of fixed points of the algorithm is characterized in several cases. Convergence criteria of the algorithm in terms of general fixed point operators are established. When applying to nonconvex feasibility including the inconsistent case, we prove local linear convergence results under mild assumptions on regularity of individual sets and of the collection of sets which need not intersect. In this special case, we refine known linear convergence criteria for the Douglas-Rachford algorithm (DR). As a consequence, for feasibility with one of the sets being affine, we establish criteria for linear and sublinear convergence of convex combinations of the alternating projection and the DR methods. These results seem to be new. We also demonstrate the seemingly improved numerical performance of this algorithm compared to the RAAR algorithm for both consistent and inconsistent sparse feasibility problems.
1 citations
TL;DR: A mirror descent (or BRegman gradient descent) algorithm based on a wisely chosen Bregman divergence is proposed, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized.
Abstract: In this paper, we consider the problem of phase retrieval, which consists of recovering an $n$-dimensional real vector from the magnitude of its $m$ linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing to remove the classical global Lipschitz continuity requirement on the gradient of the non-convex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the \iid standard Gaussian and those obtained by multiple structured illuminations through Coded Diffraction Patterns (CDP). For the Gaussian case, we show that when the number of measurements $m$ is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behaviour with a dimension-independent convergence rate. Our theoretical results are finally illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.
26 May 2022
TL;DR: In this article , the authors consider the Alexandrov method of alternating projections for finding a point in the intersection of two closed sets and highlight the two key geometric ingredients in a standard intuitive analysis of local linear convergence.
Abstract: We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the sense of Alexandrov), we are able to highlight the two key geometric ingredients in a standard intuitive analysis of local linear convergence. The first is a transversality-like condition on the intersection; the second is a convexity-like condition on one set: "uniform approximation by geodesics."
Posted Content•
TL;DR: In this paper, 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.
15 Apr 2018
TL;DR: Two projection-based techniques to compute feasible points of non-convex QCQPs with low computational complexity footprints are introduced: the first employs successive projection mappings, while the second one builds on a composition of successive and averaged projection steps.
Abstract: Quadratically constrained quadratic programming (QCQP) forms an important class of optimization tasks in various engineering disciplines Fast identification of a feasible point under low computational complexity load is critical for several approximation techniques which have been developed to solve non-convex QCQPs This paper introduces two projection-based techniques to compute feasible points of non-convex QCQPs with low computational complexity footprints: The first one employs successive projection mappings, while the second one builds on a composition of successive and averaged projection steps Extensive experiments on synthetically generated instances of non-convex quadratically constrained feasibility problems demonstrate that the simple successive-projection based technique compares favorably against state-of-the-art feasible point pursuit methods which capitalize on successive convex approximation, parallel projections and computationally demanding interior-point techniques
References
More filters
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