Convergence rate analysis for the higher order power method in best rank one approximations of tensors
Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or subli...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Hu, Shenglong [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Systematik: |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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| Übergeordnetes Werk: |
Enthalten in: Numerische Mathematik - Springer Berlin Heidelberg, 1959, 140(2018), 4 vom: 17. Juli, Seite 993-1031 |
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| Übergeordnetes Werk: |
volume:140 ; year:2018 ; number:4 ; day:17 ; month:07 ; pages:993-1031 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00211-018-0981-3 |
|---|
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OLC2040005463 |
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| 520 | |a Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. | ||
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10.1007/s00211-018-0981-3 doi (DE-627)OLC2040005463 (DE-He213)s00211-018-0981-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7310 VZ rvk Hu, Shenglong verfasserin aut Convergence rate analysis for the higher order power method in best rank one approximations of tensors 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. Li, Guoyin aut Enthalten in Numerische Mathematik Springer Berlin Heidelberg, 1959 140(2018), 4 vom: 17. Juli, Seite 993-1031 (DE-627)129081469 (DE-600)3460-5 (DE-576)014414333 0029-599X nnns volume:140 year:2018 number:4 day:17 month:07 pages:993-1031 https://doi.org/10.1007/s00211-018-0981-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_215 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4323 SA 7310 AR 140 2018 4 17 07 993-1031 |
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10.1007/s00211-018-0981-3 doi (DE-627)OLC2040005463 (DE-He213)s00211-018-0981-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7310 VZ rvk Hu, Shenglong verfasserin aut Convergence rate analysis for the higher order power method in best rank one approximations of tensors 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. Li, Guoyin aut Enthalten in Numerische Mathematik Springer Berlin Heidelberg, 1959 140(2018), 4 vom: 17. Juli, Seite 993-1031 (DE-627)129081469 (DE-600)3460-5 (DE-576)014414333 0029-599X nnns volume:140 year:2018 number:4 day:17 month:07 pages:993-1031 https://doi.org/10.1007/s00211-018-0981-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_215 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4323 SA 7310 AR 140 2018 4 17 07 993-1031 |
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10.1007/s00211-018-0981-3 doi (DE-627)OLC2040005463 (DE-He213)s00211-018-0981-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7310 VZ rvk Hu, Shenglong verfasserin aut Convergence rate analysis for the higher order power method in best rank one approximations of tensors 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. Li, Guoyin aut Enthalten in Numerische Mathematik Springer Berlin Heidelberg, 1959 140(2018), 4 vom: 17. Juli, Seite 993-1031 (DE-627)129081469 (DE-600)3460-5 (DE-576)014414333 0029-599X nnns volume:140 year:2018 number:4 day:17 month:07 pages:993-1031 https://doi.org/10.1007/s00211-018-0981-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_215 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4323 SA 7310 AR 140 2018 4 17 07 993-1031 |
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10.1007/s00211-018-0981-3 doi (DE-627)OLC2040005463 (DE-He213)s00211-018-0981-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 7310 VZ rvk Hu, Shenglong verfasserin aut Convergence rate analysis for the higher order power method in best rank one approximations of tensors 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. Li, Guoyin aut Enthalten in Numerische Mathematik Springer Berlin Heidelberg, 1959 140(2018), 4 vom: 17. Juli, Seite 993-1031 (DE-627)129081469 (DE-600)3460-5 (DE-576)014414333 0029-599X nnns volume:140 year:2018 number:4 day:17 month:07 pages:993-1031 https://doi.org/10.1007/s00211-018-0981-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_215 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4311 GBV_ILN_4323 SA 7310 AR 140 2018 4 17 07 993-1031 |
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Convergence rate analysis for the higher order power method in best rank one approximations of tensors |
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Convergence rate analysis for the higher order power method in best rank one approximations of tensors |
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Hu, Shenglong |
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Numerische Mathematik |
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Numerische Mathematik |
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Hu, Shenglong Li, Guoyin |
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10.1007/s00211-018-0981-3 |
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convergence rate analysis for the higher order power method in best rank one approximations of tensors |
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Convergence rate analysis for the higher order power method in best rank one approximations of tensors |
| abstract |
Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
| abstractGer |
Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
| abstract_unstemmed |
Abstract A popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method (HOPM). It is known in the literature that the iterative sequence generated by HOPM converges globally, while the convergence rate can be superlinear, linear or sublinear. In this paper, we examine the convergence rate of HOPM in solving the best rank one approximation problem of real tensors. We first show that the iterative sequence of HOPM always converges globally and provide an explicit eventual sublinear convergence rate. The sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space. Then, we examine the concept of nondegenerate singular vector tuples and show that, if the sequence of HOPM converges to a nondegenerate singular vector tuple, then the global convergence rate is R-linear. We show that, for almost all tensors (in the sense of Lebesgue measure), all the singular vector tuples are nondegenerate, and so, the HOPM “typically” exhibits global R-linear convergence rate. Moreover, without any regularity assumption, we establish that the sequence generated by HOPM always converges globally and R-linearly for orthogonally decomposable tensors with order at least 3. We achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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Convergence rate analysis for the higher order power method in best rank one approximations of tensors |
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