Computing sparse representations of multidimensional signals using Kronecker bases

Autores
Caiafa, César Federico; Cichocki, Andrzej
Año de publicación
2013
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case of one-dimensional signals (vectors) which involves to find the sparsest solution of an underdetermined linear system of algebraic equations. In this paper, we generalize the theory of sparse representations of vectors tomultiway arrays (tensors), i.e. signals with a multidimensional structure, by using the Tucker model. Thus, the problem is reduced to solve a large-scale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm called Kronecker-OMP as a generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for vectors. We also introduce the concept of multiway block-sparse representation of N-way arrays and develop a new greedy algorithm that exploits not only the Kronecker structure but also block-sparsity. This allows us to derive a very fast and memory efficient algorithm called N-BOMP (N-way Block OMP). We theoretically demonstrate that, under the block-sparsity assumption, our N-BOMP algorithm has not only a considerably lower complexity but it is also more precise than the classical OMP algorithm. Moreover, our algorithms can be used for very large-scale problems which are intractable by using standard approaches. We provide several simulations illustrating our results and comparing our algorithms to classical algorithms such as OMP and BP (Basis Pursuit) algorithms. We also apply the N-BOMP algorithm as a fast solution for the Compressed Sensing (CS) problem with large-scale datasets, in particular for 2D Compressive Imaging (CI) and 3D Hyperspectral CI and we show examples with real world multidimensional signals.
Fil: Caiafa, César Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico La Plata. Instituto Argentino de Radioastronomia (i); Argentina
Fil: Cichocki, Andrzej. No especifíca;
Materia
Compressed Sensing
Greedy Algorithms
Large Datasets
Multiway Arrays (Tensors)
Sparse Representations
Tucker Model
Undeterminated Linear Systems
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/4629

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network_name_str CONICET Digital (CONICET)
spelling Computing sparse representations of multidimensional signals using Kronecker basesCaiafa, César FedericoCichocki, AndrzejCompressed SensingGreedy AlgorithmsLarge DatasetsMultiway Arrays (Tensors)Sparse RepresentationsTucker ModelUndeterminated Linear Systemshttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case of one-dimensional signals (vectors) which involves to find the sparsest solution of an underdetermined linear system of algebraic equations. In this paper, we generalize the theory of sparse representations of vectors tomultiway arrays (tensors), i.e. signals with a multidimensional structure, by using the Tucker model. Thus, the problem is reduced to solve a large-scale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm called Kronecker-OMP as a generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for vectors. We also introduce the concept of multiway block-sparse representation of N-way arrays and develop a new greedy algorithm that exploits not only the Kronecker structure but also block-sparsity. This allows us to derive a very fast and memory efficient algorithm called N-BOMP (N-way Block OMP). We theoretically demonstrate that, under the block-sparsity assumption, our N-BOMP algorithm has not only a considerably lower complexity but it is also more precise than the classical OMP algorithm. Moreover, our algorithms can be used for very large-scale problems which are intractable by using standard approaches. We provide several simulations illustrating our results and comparing our algorithms to classical algorithms such as OMP and BP (Basis Pursuit) algorithms. We also apply the N-BOMP algorithm as a fast solution for the Compressed Sensing (CS) problem with large-scale datasets, in particular for 2D Compressive Imaging (CI) and 3D Hyperspectral CI and we show examples with real world multidimensional signals.Fil: Caiafa, César Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico La Plata. Instituto Argentino de Radioastronomia (i); ArgentinaFil: Cichocki, Andrzej. No especifíca;M I T Press2013-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/4629Caiafa, César Federico; Cichocki, Andrzej ; Computing sparse representations of multidimensional signals using Kronecker bases; M I T Press; Neural Computation; 25; 1; 1-2013; 186-2200899-7667enginfo:eu-repo/semantics/altIdentifier/doi/10.1162/NECO_a_00385info:eu-repo/semantics/altIdentifier/url/http://www.mitpressjournals.org/doi/abs/10.1162/NECO_a_00385info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:04:05Zoai:ri.conicet.gov.ar:11336/4629instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 10:04:05.751CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Computing sparse representations of multidimensional signals using Kronecker bases
title Computing sparse representations of multidimensional signals using Kronecker bases
spellingShingle Computing sparse representations of multidimensional signals using Kronecker bases
Caiafa, César Federico
Compressed Sensing
Greedy Algorithms
Large Datasets
Multiway Arrays (Tensors)
Sparse Representations
Tucker Model
Undeterminated Linear Systems
title_short Computing sparse representations of multidimensional signals using Kronecker bases
title_full Computing sparse representations of multidimensional signals using Kronecker bases
title_fullStr Computing sparse representations of multidimensional signals using Kronecker bases
title_full_unstemmed Computing sparse representations of multidimensional signals using Kronecker bases
title_sort Computing sparse representations of multidimensional signals using Kronecker bases
dc.creator.none.fl_str_mv Caiafa, César Federico
Cichocki, Andrzej
author Caiafa, César Federico
author_facet Caiafa, César Federico
Cichocki, Andrzej
author_role author
author2 Cichocki, Andrzej
author2_role author
dc.subject.none.fl_str_mv Compressed Sensing
Greedy Algorithms
Large Datasets
Multiway Arrays (Tensors)
Sparse Representations
Tucker Model
Undeterminated Linear Systems
topic Compressed Sensing
Greedy Algorithms
Large Datasets
Multiway Arrays (Tensors)
Sparse Representations
Tucker Model
Undeterminated Linear Systems
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case of one-dimensional signals (vectors) which involves to find the sparsest solution of an underdetermined linear system of algebraic equations. In this paper, we generalize the theory of sparse representations of vectors tomultiway arrays (tensors), i.e. signals with a multidimensional structure, by using the Tucker model. Thus, the problem is reduced to solve a large-scale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm called Kronecker-OMP as a generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for vectors. We also introduce the concept of multiway block-sparse representation of N-way arrays and develop a new greedy algorithm that exploits not only the Kronecker structure but also block-sparsity. This allows us to derive a very fast and memory efficient algorithm called N-BOMP (N-way Block OMP). We theoretically demonstrate that, under the block-sparsity assumption, our N-BOMP algorithm has not only a considerably lower complexity but it is also more precise than the classical OMP algorithm. Moreover, our algorithms can be used for very large-scale problems which are intractable by using standard approaches. We provide several simulations illustrating our results and comparing our algorithms to classical algorithms such as OMP and BP (Basis Pursuit) algorithms. We also apply the N-BOMP algorithm as a fast solution for the Compressed Sensing (CS) problem with large-scale datasets, in particular for 2D Compressive Imaging (CI) and 3D Hyperspectral CI and we show examples with real world multidimensional signals.
Fil: Caiafa, César Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico La Plata. Instituto Argentino de Radioastronomia (i); Argentina
Fil: Cichocki, Andrzej. No especifíca;
description Recently, there is a great interest in sparse representations of signals under the assumption that signals (datasets) can be well approximated by a linear combination of few elements of a known basis (dictionary). Many algorithms have been developed to find such kind of representations for the case of one-dimensional signals (vectors) which involves to find the sparsest solution of an underdetermined linear system of algebraic equations. In this paper, we generalize the theory of sparse representations of vectors tomultiway arrays (tensors), i.e. signals with a multidimensional structure, by using the Tucker model. Thus, the problem is reduced to solve a large-scale underdetermined linear system of equations possessing a Kronecker structure, for which we have developed a greedy algorithm called Kronecker-OMP as a generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for vectors. We also introduce the concept of multiway block-sparse representation of N-way arrays and develop a new greedy algorithm that exploits not only the Kronecker structure but also block-sparsity. This allows us to derive a very fast and memory efficient algorithm called N-BOMP (N-way Block OMP). We theoretically demonstrate that, under the block-sparsity assumption, our N-BOMP algorithm has not only a considerably lower complexity but it is also more precise than the classical OMP algorithm. Moreover, our algorithms can be used for very large-scale problems which are intractable by using standard approaches. We provide several simulations illustrating our results and comparing our algorithms to classical algorithms such as OMP and BP (Basis Pursuit) algorithms. We also apply the N-BOMP algorithm as a fast solution for the Compressed Sensing (CS) problem with large-scale datasets, in particular for 2D Compressive Imaging (CI) and 3D Hyperspectral CI and we show examples with real world multidimensional signals.
publishDate 2013
dc.date.none.fl_str_mv 2013-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/4629
Caiafa, César Federico; Cichocki, Andrzej ; Computing sparse representations of multidimensional signals using Kronecker bases; M I T Press; Neural Computation; 25; 1; 1-2013; 186-220
0899-7667
url http://hdl.handle.net/11336/4629
identifier_str_mv Caiafa, César Federico; Cichocki, Andrzej ; Computing sparse representations of multidimensional signals using Kronecker bases; M I T Press; Neural Computation; 25; 1; 1-2013; 186-220
0899-7667
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1162/NECO_a_00385
info:eu-repo/semantics/altIdentifier/url/http://www.mitpressjournals.org/doi/abs/10.1162/NECO_a_00385
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv M I T Press
publisher.none.fl_str_mv M I T Press
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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