Projection Methods for Uniformly Convex Expandable Sets

Projection Methods for Uniformly Convex Expandable Sets

Many problems in medical image reconstruction and machine learning can be formulated as nonconvex set theoretic feasibility problems. Among efficient methods that can be put to work in practice, successive projection algorithms have received a lot of attention in the case of convex constraint sets....

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Main Authors: Stéphane Chrétien, Pascal Bondon
Format: Article
Language:English
Published: MDPI AG 2020-07-01
Series:Mathematics
Subjects:
cyclic projections
nonconvex sets
uniformly convex sets
strong convergence
Online Access:https://www.mdpi.com/2227-7390/8/7/1108
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spelling doaj-49a004ce9382433fae77026a7be5109a2020-11-25T03:55:15ZengMDPI AGMathematics2227-73902020-07-0181108110810.3390/math8071108Projection Methods for Uniformly Convex Expandable SetsStéphane Chrétien0Pascal Bondon1Laboratoire ERIC, Université Lyon 2, 69500 Bron, FranceLaboratoire des Signaux et Systèmes, CentraleSupélec, CNRS, Université Paris-Saclay, 91190 Gif-sur-Yvette, FranceMany problems in medical image reconstruction and machine learning can be formulated as nonconvex set theoretic feasibility problems. Among efficient methods that can be put to work in practice, successive projection algorithms have received a lot of attention in the case of convex constraint sets. In the present work, we provide a theoretical study of a general projection method in the case where the constraint sets are nonconvex and satisfy some other structural properties. We apply our algorithm to image recovery in magnetic resonance imaging (MRI) and to a signal denoising in the spirit of Cadzow’s method.https://www.mdpi.com/2227-7390/8/7/1108cyclic projectionsnonconvex setsuniformly convex setsstrong convergence
collection DOAJ
language English
format Article
sources DOAJ
author Stéphane Chrétien
Pascal Bondon
spellingShingle Stéphane Chrétien
Pascal Bondon
Projection Methods for Uniformly Convex Expandable Sets
Mathematics
cyclic projections
nonconvex sets
uniformly convex sets
strong convergence
author_facet Stéphane Chrétien
Pascal Bondon
author_sort Stéphane Chrétien
title Projection Methods for Uniformly Convex Expandable Sets
title_short Projection Methods for Uniformly Convex Expandable Sets
title_full Projection Methods for Uniformly Convex Expandable Sets
title_fullStr Projection Methods for Uniformly Convex Expandable Sets
title_full_unstemmed Projection Methods for Uniformly Convex Expandable Sets
title_sort projection methods for uniformly convex expandable sets
publisher MDPI AG
series Mathematics
issn 2227-7390
publishDate 2020-07-01
description Many problems in medical image reconstruction and machine learning can be formulated as nonconvex set theoretic feasibility problems. Among efficient methods that can be put to work in practice, successive projection algorithms have received a lot of attention in the case of convex constraint sets. In the present work, we provide a theoretical study of a general projection method in the case where the constraint sets are nonconvex and satisfy some other structural properties. We apply our algorithm to image recovery in magnetic resonance imaging (MRI) and to a signal denoising in the spirit of Cadzow’s method.
topic cyclic projections
nonconvex sets
uniformly convex sets
strong convergence
url https://www.mdpi.com/2227-7390/8/7/1108
work_keys_str_mv AT stephanechretien projectionmethodsforuniformlyconvexexpandablesets
AT pascalbondon projectionmethodsforuniformlyconvexexpandablesets
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