Singular value decompositions and low rank approximations of multi-linear functionals

F. Belzen, van; S. Weiland; J. Graaf, de

doi:10.1109/CDC.2007.4434697

Singular value decompositions and low rank approximations of multi-linear functionals

F. Belzen, van
, S. Weiland
, J. Graaf, de

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

10 Citations (Scopus)

4 Downloads (Pure)

Abstract

The singular value decomposition is among the most important algebraic tools for solving many approximation problems in model reduction, data compression, system identification and signal processing. Nevertheless, there is no straightforward generalization of the algebraic concept of singular values and singular value decompositions to multi-linear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper introduces a notion of singular values for arbitrary multi-linear mappings. An upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.

Original language	English
Title of host publication	Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA
Place of Publication	Piscataway, New Jersey, USA
Publisher	Institute of Electrical and Electronics Engineers
Pages	3751-3756
ISBN (Print)	978-1-424-41497-0
DOIs	https://doi.org/10.1109/CDC.2007.4434697
Publication status	Published - 2007
Event	46th IEEE Conference on Decision and Control (CDC 2007) - New Orleans, United States Duration: 12 Dec 2007 → 14 Dec 2007 Conference number: 46

Conference

Conference	46th IEEE Conference on Decision and Control (CDC 2007)
Abbreviated title	CDC 2007
Country/Territory	United States
City	New Orleans
Period	12/12/07 → 14/12/07

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10.1109/CDC.2007.4434697

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Belzen, van, F., Weiland, S., & Graaf, de, J. (2007). Singular value decompositions and low rank approximations of multi-linear functionals. In Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA (pp. 3751-3756). Institute of Electrical and Electronics Engineers. https://doi.org/10.1109/CDC.2007.4434697

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title = "Singular value decompositions and low rank approximations of multi-linear functionals",

abstract = "The singular value decomposition is among the most important algebraic tools for solving many approximation problems in model reduction, data compression, system identification and signal processing. Nevertheless, there is no straightforward generalization of the algebraic concept of singular values and singular value decompositions to multi-linear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper introduces a notion of singular values for arbitrary multi-linear mappings. An upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.",

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booktitle = "Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA",

publisher = "Institute of Electrical and Electronics Engineers",

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note = "46th IEEE Conference on Decision and Control (CDC 2007), CDC 2007 ; Conference date: 12-12-2007 Through 14-12-2007",

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Belzen, van, F, Weiland, S & Graaf, de, J 2007, Singular value decompositions and low rank approximations of multi-linear functionals. in Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, USA, pp. 3751-3756, 46th IEEE Conference on Decision and Control (CDC 2007), New Orleans, Louisiana, United States, 12/12/07. https://doi.org/10.1109/CDC.2007.4434697

Singular value decompositions and low rank approximations of multi-linear functionals. / Belzen, van, F.; Weiland, S.; Graaf, de, J.
Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA. Piscataway, New Jersey, USA: Institute of Electrical and Electronics Engineers, 2007. p. 3751-3756.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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N2 - The singular value decomposition is among the most important algebraic tools for solving many approximation problems in model reduction, data compression, system identification and signal processing. Nevertheless, there is no straightforward generalization of the algebraic concept of singular values and singular value decompositions to multi-linear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper introduces a notion of singular values for arbitrary multi-linear mappings. An upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.

AB - The singular value decomposition is among the most important algebraic tools for solving many approximation problems in model reduction, data compression, system identification and signal processing. Nevertheless, there is no straightforward generalization of the algebraic concept of singular values and singular value decompositions to multi-linear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper introduces a notion of singular values for arbitrary multi-linear mappings. An upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.

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BT - Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA

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T2 - 46th IEEE Conference on Decision and Control (CDC 2007)

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ER -

Belzen, van F, Weiland S, Graaf, de J. Singular value decompositions and low rank approximations of multi-linear functionals. In Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana, USA. Piscataway, New Jersey, USA: Institute of Electrical and Electronics Engineers. 2007. p. 3751-3756 doi: 10.1109/CDC.2007.4434697

Singular value decompositions and low rank approximations of multi-linear functionals

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