First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD

Emilio Rafael Balda; Sher Ali Cheema; Jens Steinwandt; Martin Haardt; Amir Weiss; Arie Yeredor

doi:10.1109/acssc.2016.7869677

First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD

Emilio Rafael Balda, Sher Ali Cheema, Jens Steinwandt, Martin Haardt, Amir Weiss, Arie Yeredor

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

7 Scopus citations

Abstract

The truncated version of the higher-order singular value decomposition (HOSVD) has a great significance in multi-dimensional tensor-based signal processing. It allows to extract the principal components from noisy observations in order to find a low-rank approximation of the multi-dimensional data. In this paper, we address the question of how good the approximation is by analytically quantifying the tensor reconstruction error introduced by the truncated HOSVD. We present a first-order perturbation analysis of the truncated HOSVD to obtain analytical expressions for the signal subspace error in each dimension as well as the tensor reconstruction error induced by the low-rank approximation of the noise corrupted tensor. The results are asymptotic in the signal-to-noise ratio (SNR) and expressed in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Empirical simulation results verify the obtained analytical expressions.

Original language	English
Title of host publication	Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016
Editors	Michael B. Matthews
Publisher	IEEE Computer Society
Pages	1723-1727
Number of pages	5
ISBN (Electronic)	9781538639542
DOIs	https://doi.org/10.1109/acssc.2016.7869677
State	Published - 1 Mar 2017
Externally published	Yes
Event	50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016 - Pacific Grove, United States Duration: 6 Nov 2016 → 9 Nov 2016

Publication series

Name	Conference Record - Asilomar Conference on Signals, Systems and Computers
ISSN (Print)	1058-6393

Conference

Conference	50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016
Country/Territory	United States
City	Pacific Grove
Period	6/11/16 → 9/11/16

Bibliographical note

Publisher Copyright:
© 2016 IEEE.

Keywords

higher-order singular value decomposition (HOSVD)
Perturbation analysis
tensor signal processing

Access to Document

10.1109/acssc.2016.7869677

Cite this

Balda, E. R., Cheema, S. A., Steinwandt, J., Haardt, M., Weiss, A., & Yeredor, A. (2017). First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD. In M. B. Matthews (Ed.), Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016 (pp. 1723-1727). Article 7869677 (Conference Record - Asilomar Conference on Signals, Systems and Computers). IEEE Computer Society. https://doi.org/10.1109/acssc.2016.7869677

Balda, Emilio Rafael ; Cheema, Sher Ali ; Steinwandt, Jens et al. / First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD. Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016. editor / Michael B. Matthews. IEEE Computer Society, 2017. pp. 1723-1727 (Conference Record - Asilomar Conference on Signals, Systems and Computers).

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abstract = "The truncated version of the higher-order singular value decomposition (HOSVD) has a great significance in multi-dimensional tensor-based signal processing. It allows to extract the principal components from noisy observations in order to find a low-rank approximation of the multi-dimensional data. In this paper, we address the question of how good the approximation is by analytically quantifying the tensor reconstruction error introduced by the truncated HOSVD. We present a first-order perturbation analysis of the truncated HOSVD to obtain analytical expressions for the signal subspace error in each dimension as well as the tensor reconstruction error induced by the low-rank approximation of the noise corrupted tensor. The results are asymptotic in the signal-to-noise ratio (SNR) and expressed in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Empirical simulation results verify the obtained analytical expressions.",

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Balda, ER, Cheema, SA, Steinwandt, J, Haardt, M, Weiss, A & Yeredor, A 2017, First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD. in MB Matthews (ed.), Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016., 7869677, Conference Record - Asilomar Conference on Signals, Systems and Computers, IEEE Computer Society, pp. 1723-1727, 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016, Pacific Grove, United States, 6/11/16. https://doi.org/10.1109/acssc.2016.7869677

First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD. / Balda, Emilio Rafael; Cheema, Sher Ali; Steinwandt, Jens et al.
Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016. ed. / Michael B. Matthews. IEEE Computer Society, 2017. p. 1723-1727 7869677 (Conference Record - Asilomar Conference on Signals, Systems and Computers).

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

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AU - Cheema, Sher Ali

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AU - Weiss, Amir

AU - Yeredor, Arie

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N2 - The truncated version of the higher-order singular value decomposition (HOSVD) has a great significance in multi-dimensional tensor-based signal processing. It allows to extract the principal components from noisy observations in order to find a low-rank approximation of the multi-dimensional data. In this paper, we address the question of how good the approximation is by analytically quantifying the tensor reconstruction error introduced by the truncated HOSVD. We present a first-order perturbation analysis of the truncated HOSVD to obtain analytical expressions for the signal subspace error in each dimension as well as the tensor reconstruction error induced by the low-rank approximation of the noise corrupted tensor. The results are asymptotic in the signal-to-noise ratio (SNR) and expressed in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Empirical simulation results verify the obtained analytical expressions.

AB - The truncated version of the higher-order singular value decomposition (HOSVD) has a great significance in multi-dimensional tensor-based signal processing. It allows to extract the principal components from noisy observations in order to find a low-rank approximation of the multi-dimensional data. In this paper, we address the question of how good the approximation is by analytically quantifying the tensor reconstruction error introduced by the truncated HOSVD. We present a first-order perturbation analysis of the truncated HOSVD to obtain analytical expressions for the signal subspace error in each dimension as well as the tensor reconstruction error induced by the low-rank approximation of the noise corrupted tensor. The results are asymptotic in the signal-to-noise ratio (SNR) and expressed in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Empirical simulation results verify the obtained analytical expressions.

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Balda ER, Cheema SA, Steinwandt J, Haardt M, Weiss A, Yeredor A. First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD. In Matthews MB, editor, Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016. IEEE Computer Society. 2017. p. 1723-1727. 7869677. (Conference Record - Asilomar Conference on Signals, Systems and Computers). doi: 10.1109/acssc.2016.7869677

First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD

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