Alternating projections on nontangential manifolds
type: Item
instance of: scholarly article
title: Alternating projections on nontangential manifolds
zbMATH Open document ID: 1283.41026
author: Fredrik Andersson
author: Marcus Carlsson
published in: Constructive Approximation
publication date: 2013-12-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.4055
review text: Let \(\mathcal M_1,\mathcal M_2\) be manifolds in a finite-dimensional real Hilbert space \(\mathcal K\), and \(\mathcal M=\mathcal M_1\cap\mathcal M_2\). Denote by \(\pi_1\), \(\pi_2\) and \(\pi\) the corresponding metric projections. Given a point \(B\in \mathcal K\), the sequence \((B_k)\) of alternating projections is defined by \(B_0=B,\, B_{k+1}=\pi_1(B_k)\) if \(k\) is even, and \(B_{k+1}=\pi_2(B_k)\) if \(k\) is odd. A classical result of von Neumann (1949) asserts that if \(\mathcal M_1\), \(\mathcal M_2\) are affine linear manifolds, then \((B_k)\) converges to \(\pi(B)\). The authors study the convergence of this sequence in the case of nonlinear manifolds. The key notion used in the paper is that of nontangential points. A point \(A\in \mathcal M_1\cap\mathcal M_2\) is called nontangential if \(T_{\mathcal M_1}(A)\cap T_{\mathcal M_2}(A)=T_{\mathcal M_1\cap\mathcal M_2}(A)\) (where by \(T_{\mathcal N}(A)\) one denotes the tangent space to a manifold \(\mathcal N\) at a point \(A\in \mathcal N),\) a condition that can be also expressed in terms of the positivity of the angle \(\sigma(A)\) between \(\mathcal M_1,\mathcal M_2\) at \(A\). A point \(A\) is called transversal if \(T_{\mathcal M_1}(A)+ T_{\mathcal M_2}(A)=\mathcal K\). Since any transversal point is nontangential, the results obtained in the present paper extend those obtained by \textit{A. S. Lewis} and \textit{J. Malick} [Math. Oper. Res. 33, No. 1, 216--234 (2008; Zbl 1163.65040)] for transversal points. The main result of the paper (Theorem 5.1) asserts that if \(\mathcal M_1\), \(\mathcal M_2\) are \(C^2\)-manifolds and \(A\in \mathcal M_1\cap\mathcal M_2\) is a nontangential point, then for every \(\varepsilon >0\) and \( 1>c>\sigma(A),\) there exists \(r>0\) such that for any \(B\in\mathcal B(A,r)\) (the open ball) the corresponding sequence \((B_k)\) of alternating projections converges to a point \(B_\infty\in \mathcal M_1\cap\mathcal M_2 ,\; \|B_\infty-\pi(B)\|\leq\varepsilon \|B-\pi(…
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47N10
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 6242066
zbMATH Keywords: metric projection
zbMATH Keywords: alternating projections
zbMATH Keywords: algorithms
zbMATH Keywords: rate of convergence
zbMATH Keywords: smooth manifold
zbMATH Keywords: low-rank approximation
zbMATH Keywords: real algebraic variety
reviewed by: Stefan Cobzaş
describes a project that uses: SDPT3
MaRDI profile type: Publication
OpenAlex ID: W2074065373
arXiv ID: 1107.4055
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DOI: 10.1007/S00365-013-9213-3
description: scientific article; zbMATH DE number 6242066