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

School of Electrical Engineering

Tel Aviv University

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

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
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
Volume	0

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

Keywords

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

ASJC Scopus subject areas

Signal Processing
Computer Networks and Communications

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; Vol. 0). 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).

@inproceedings{d238920ff6bc4489b1d3eb15e2fdb9e6,

title = "First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD",

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.",

keywords = "Perturbation analysis, higher-order singular value decomposition (HOSVD), tensor signal processing",

author = "Balda, \{Emilio Rafael\} and Cheema, \{Sher Ali\} and Jens Steinwandt and Martin Haardt and Amir Weiss and Arie Yeredor",

note = "Publisher Copyright: {\textcopyright} 2016 IEEE.; 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016 ; Conference date: 06-11-2016 Through 09-11-2016",

year = "2017",

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doi = "10.1109/acssc.2016.7869677",

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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, vol. 0, 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; Vol. 0).

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

TY - GEN

T1 - First-order perturbation analysis of low-rank tensor approximations based on the truncated HOSVD

AU - Balda, Emilio Rafael

AU - Cheema, Sher Ali

AU - Steinwandt, Jens

AU - Haardt, Martin

AU - Weiss, Amir

AU - Yeredor, Arie

PY - 2017/3/1

Y1 - 2017/3/1

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.

KW - Perturbation analysis

KW - higher-order singular value decomposition (HOSVD)

KW - tensor signal processing

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DO - 10.1109/acssc.2016.7869677

M3 - Conference contribution

T3 - Conference Record - Asilomar Conference on Signals, Systems and Computers

SP - 1723

EP - 1727

BT - Conference Record of the 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016

A2 - Matthews, Michael B.

PB - IEEE Computer Society

T2 - 50th Asilomar Conference on Signals, Systems and Computers, ACSSC 2016

Y2 - 6 November 2016 through 9 November 2016

ER -

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

Abstract

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Keywords

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