Constructive Approximation

, Volume 38, Issue 3, pp 489–525 | Cite as

Alternating Projections on Nontangential Manifolds

Article

Abstract

We consider sequences \((B_{k})_{k=0}^{\infty}\) of points obtained by projecting a given point B=B0 back and forth between two manifolds \(\mathcal{M}_{1}\) and \(\mathcal{M}_{2}\), and give conditions guaranteeing that the sequence converges to a limit \(B_{\infty}\in\mathcal{M}_{1}\cap\mathcal{M}_{2}\). Our motivation is the study of algorithms based on finding the limit of such sequences, which have proved useful in a number of areas. The intersection is typically a set with desirable properties but for which there is no efficient method for finding the closest point Bopt in \(\mathcal{M}_{1}\cap\mathcal{M}_{2}\). Under appropriate conditions, we prove not only that the sequence of alternating projections converges, but that the limit point is fairly close to Bopt, in a manner relative to the distance ∥B0Bopt∥, thereby significantly improving earlier results in the field.

Keywords

Alternating projections Convergence Non-convexity Low-rank approximation Manifolds 

Mathematics Subject Classification

41A65 49Q99 53B25 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

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