Stable, robust and super fast reconstruction of tensors using multi-way projections

Autores
Caiafa, Cesar Federico; Cichocki, Andrzej
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an Nth-order data tensor X from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order N. In addition, it is proved that, in the matrix case and in a particular case with 3rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction X is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter t = 0 > 0 exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using t = 0, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.
Fil: Caiafa, Cesar Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico la Plata. Instituto Argentino de Radioastronomia (i); Argentina
Fil: Cichocki, Andrzej. Brain Science Institute. Riken; Japón
Materia
Compressed sensing
Kronecker-CS
Low-rank approximations
Multiway analysis
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:11336/5832

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network_name_str CONICET Digital (CONICET)
spelling Stable, robust and super fast reconstruction of tensors using multi-way projectionsCaiafa, Cesar FedericoCichocki, AndrzejCompressed sensingKronecker-CSLow-rank approximationsMultiway analysishttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an Nth-order data tensor X from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order N. In addition, it is proved that, in the matrix case and in a particular case with 3rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction X is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter t = 0 > 0 exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using t = 0, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.Fil: Caiafa, Cesar Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico la Plata. Instituto Argentino de Radioastronomia (i); ArgentinaFil: Cichocki, Andrzej. Brain Science Institute. Riken; JapónInstitute of Electrical and Electronics Engineers2015-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/5832Caiafa, Cesar Federico; Cichocki, Andrzej; Stable, robust and super fast reconstruction of tensors using multi-way projections; Institute of Electrical and Electronics Engineers; IEEE Transactions On Signal Processing; 63; 3; 1-2015; 780-7931053-587Xenginfo:eu-repo/semantics/altIdentifier/doi/10.1109/TSP.2014.2385040info:eu-repo/semantics/altIdentifier/url/http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6994852info:eu-repo/semantics/altIdentifier/doi/info:eu-repo/semantics/altIdentifier/url/http://arxiv.org/abs/1406.3295v2info:eu-repo/semantics/altIdentifier/arxiv/1406.3295v2info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:58:05Zoai:ri.conicet.gov.ar:11336/5832instacron: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 09:58:06.073CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Stable, robust and super fast reconstruction of tensors using multi-way projections
title Stable, robust and super fast reconstruction of tensors using multi-way projections
spellingShingle Stable, robust and super fast reconstruction of tensors using multi-way projections
Caiafa, Cesar Federico
Compressed sensing
Kronecker-CS
Low-rank approximations
Multiway analysis
title_short Stable, robust and super fast reconstruction of tensors using multi-way projections
title_full Stable, robust and super fast reconstruction of tensors using multi-way projections
title_fullStr Stable, robust and super fast reconstruction of tensors using multi-way projections
title_full_unstemmed Stable, robust and super fast reconstruction of tensors using multi-way projections
title_sort Stable, robust and super fast reconstruction of tensors using multi-way projections
dc.creator.none.fl_str_mv Caiafa, Cesar Federico
Cichocki, Andrzej
author Caiafa, Cesar Federico
author_facet Caiafa, Cesar Federico
Cichocki, Andrzej
author_role author
author2 Cichocki, Andrzej
author2_role author
dc.subject.none.fl_str_mv Compressed sensing
Kronecker-CS
Low-rank approximations
Multiway analysis
topic Compressed sensing
Kronecker-CS
Low-rank approximations
Multiway analysis
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an Nth-order data tensor X from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order N. In addition, it is proved that, in the matrix case and in a particular case with 3rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction X is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter t = 0 > 0 exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using t = 0, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.
Fil: Caiafa, Cesar Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico la Plata. Instituto Argentino de Radioastronomia (i); Argentina
Fil: Cichocki, Andrzej. Brain Science Institute. Riken; Japón
description In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an Nth-order data tensor X from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order N. In addition, it is proved that, in the matrix case and in a particular case with 3rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction X is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter t = 0 > 0 exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using t = 0, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets.
publishDate 2015
dc.date.none.fl_str_mv 2015-01
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/11336/5832
Caiafa, Cesar Federico; Cichocki, Andrzej; Stable, robust and super fast reconstruction of tensors using multi-way projections; Institute of Electrical and Electronics Engineers; IEEE Transactions On Signal Processing; 63; 3; 1-2015; 780-793
1053-587X
url http://hdl.handle.net/11336/5832
identifier_str_mv Caiafa, Cesar Federico; Cichocki, Andrzej; Stable, robust and super fast reconstruction of tensors using multi-way projections; Institute of Electrical and Electronics Engineers; IEEE Transactions On Signal Processing; 63; 3; 1-2015; 780-793
1053-587X
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1109/TSP.2014.2385040
info:eu-repo/semantics/altIdentifier/url/http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6994852
info:eu-repo/semantics/altIdentifier/doi/
info:eu-repo/semantics/altIdentifier/url/http://arxiv.org/abs/1406.3295v2
info:eu-repo/semantics/altIdentifier/arxiv/1406.3295v2
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
application/pdf
dc.publisher.none.fl_str_mv Institute of Electrical and Electronics Engineers
publisher.none.fl_str_mv Institute of Electrical and Electronics Engineers
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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