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- Projections_onto_convex_sets abstract "In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. The simplest case, when the sets are affine spaces, was analyzed by John von Neumann. The case when the sets are affine spaces is special, since the iterates not only converge to a point in the intersection (assuming the intersection is non-empty) but in fact to the orthogonal projection onto the intersection of the initial iterate. For general closed convex sets, the limit point need not be the projection. Classical work on the case of two closed convex sets shows that the rate of convergence of the iterates is linear.There are now extensions that consider cases when there are more than one set, or when the sets are not convex, or that give faster convergence rates. Analysis of POCS and related methods attempt to show that the algorithm converges (and if so, find the rate of convergence), and whether it converges to the projection of the original point. These questions are largely known for simple cases, but a topic of active research for the extensions. There are also variants of the algorithm, such as Dykstra's projection algorithm. See the references in the further reading section for an overview of the variants, extensions and applications of the POCS method; a good historical background can be found in section III of.".
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- Projections_onto_convex_sets wikiPageWikiLink Affine_space.
- Projections_onto_convex_sets wikiPageWikiLink Affine_spaces.
- Projections_onto_convex_sets wikiPageWikiLink Category:Convex_geometry.
- Projections_onto_convex_sets wikiPageWikiLink Closed_set.
- Projections_onto_convex_sets wikiPageWikiLink Convergent_series.
- Projections_onto_convex_sets wikiPageWikiLink Convex_set.
- Projections_onto_convex_sets wikiPageWikiLink Dykstras_projection_algorithm.
- Projections_onto_convex_sets wikiPageWikiLink Intersection_(set_theory).
- Projections_onto_convex_sets wikiPageWikiLink John_von_Neumann.
- Projections_onto_convex_sets wikiPageWikiLink Projection_(linear_algebra).
- Projections_onto_convex_sets wikiPageWikiLink Rate_of_convergence.
- Projections_onto_convex_sets wikiPageWikiLink Sequence.
- Projections_onto_convex_sets wikiPageWikiLink Tensor_product.
- Projections_onto_convex_sets wikiPageWikiLink File:Projections_onto_convex_avg_sets_circles.svg.
- Projections_onto_convex_sets wikiPageWikiLink File:Projections_onto_convex_sets_circles.svg.
- Projections_onto_convex_sets wikiPageWikiLinkText "Projection Onto Convex Sets".
- Projections_onto_convex_sets wikiPageWikiLinkText "Projections onto convex sets".
- Projections_onto_convex_sets wikiPageWikiLinkText "projections onto convex sets".
- Projections_onto_convex_sets hasPhotoCollection Projections_onto_convex_sets.
- Projections_onto_convex_sets subject Category:Convex_geometry.
- Projections_onto_convex_sets hypernym Method.
- Projections_onto_convex_sets type Software.
- Projections_onto_convex_sets comment "In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. The simplest case, when the sets are affine spaces, was analyzed by John von Neumann.".
- Projections_onto_convex_sets label "Projections onto convex sets".
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- Projections_onto_convex_sets sameAs Q7249460.
- Projections_onto_convex_sets sameAs Q7249460.
- Projections_onto_convex_sets wasDerivedFrom Projections_onto_convex_sets?oldid=679097084.
- Projections_onto_convex_sets depiction Projections_onto_convex_sets_circles.svg.
- Projections_onto_convex_sets isPrimaryTopicOf Projections_onto_convex_sets.