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The *Canonical Polyadic Decomposition (CPD)* is the tensor analog of the Singular Value Decomposition (SVD) for a matrix and has many data science applications including signal processing and machine learning. For the CPD, the *Alternating Least Squares (ALS)* algorithm has been used extensively. Although the ALS algorithm is simple, it is sensitive to a noise of a data tensor in the applications. In this paper, we propose a novel strategy to realize the noise suppression for the CPD. The proposed strategy is decomposed into two steps: (Step 1) denoising the given tensor and (Step 2) solving the *exact* CPD of the denoised tensor. Step 1 can be realized by solving a *structured low-rank approximation* with the *Douglas-Rachford splitting algorithm* and then Step 2 can be realized by solving the simultaneous diagonalization of a matrix tuple constructed by the denoised tensor with the DODO method. Numerical experiments show that the proposed algorithm works well even in typical cases where the ALS algorithm suffers from the so-called bottleneck/swamp effect.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E105-A No.1 pp.11-24

- Publication Date
- 2022/01/01

- Publicized
- 2021/06/23

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.2020EAP1138

- Type of Manuscript
- PAPER

- Category
- Digital Signal Processing

Riku AKEMA

Tokyo Institute of Technology

Masao YAMAGISHI

Tokyo Institute of Technology

Isao YAMADA

Tokyo Institute of Technology

The copyright of the original papers published on this site belongs to IEICE. Unauthorized use of the original or translated papers is prohibited. See IEICE Provisions on Copyright for details.

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Riku AKEMA, Masao YAMAGISHI, Isao YAMADA, "A Robust Canonical Polyadic Tensor Decomposition via Structured Low-Rank Matrix Approximation" in IEICE TRANSACTIONS on Fundamentals,
vol. E105-A, no. 1, pp. 11-24, January 2022, doi: 10.1587/transfun.2020EAP1138.

Abstract: The *Canonical Polyadic Decomposition (CPD)* is the tensor analog of the Singular Value Decomposition (SVD) for a matrix and has many data science applications including signal processing and machine learning. For the CPD, the *Alternating Least Squares (ALS)* algorithm has been used extensively. Although the ALS algorithm is simple, it is sensitive to a noise of a data tensor in the applications. In this paper, we propose a novel strategy to realize the noise suppression for the CPD. The proposed strategy is decomposed into two steps: (Step 1) denoising the given tensor and (Step 2) solving the *exact* CPD of the denoised tensor. Step 1 can be realized by solving a *structured low-rank approximation* with the *Douglas-Rachford splitting algorithm* and then Step 2 can be realized by solving the simultaneous diagonalization of a matrix tuple constructed by the denoised tensor with the DODO method. Numerical experiments show that the proposed algorithm works well even in typical cases where the ALS algorithm suffers from the so-called bottleneck/swamp effect.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2020EAP1138/_p

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@ARTICLE{e105-a_1_11,

author={Riku AKEMA, Masao YAMAGISHI, Isao YAMADA, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={A Robust Canonical Polyadic Tensor Decomposition via Structured Low-Rank Matrix Approximation},

year={2022},

volume={E105-A},

number={1},

pages={11-24},

abstract={The *Canonical Polyadic Decomposition (CPD)* is the tensor analog of the Singular Value Decomposition (SVD) for a matrix and has many data science applications including signal processing and machine learning. For the CPD, the *Alternating Least Squares (ALS)* algorithm has been used extensively. Although the ALS algorithm is simple, it is sensitive to a noise of a data tensor in the applications. In this paper, we propose a novel strategy to realize the noise suppression for the CPD. The proposed strategy is decomposed into two steps: (Step 1) denoising the given tensor and (Step 2) solving the *exact* CPD of the denoised tensor. Step 1 can be realized by solving a *structured low-rank approximation* with the *Douglas-Rachford splitting algorithm* and then Step 2 can be realized by solving the simultaneous diagonalization of a matrix tuple constructed by the denoised tensor with the DODO method. Numerical experiments show that the proposed algorithm works well even in typical cases where the ALS algorithm suffers from the so-called bottleneck/swamp effect.},

keywords={},

doi={10.1587/transfun.2020EAP1138},

ISSN={1745-1337},

month={January},}

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TY - JOUR

TI - A Robust Canonical Polyadic Tensor Decomposition via Structured Low-Rank Matrix Approximation

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 11

EP - 24

AU - Riku AKEMA

AU - Masao YAMAGISHI

AU - Isao YAMADA

PY - 2022

DO - 10.1587/transfun.2020EAP1138

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E105-A

IS - 1

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - January 2022

AB - The *Canonical Polyadic Decomposition (CPD)* is the tensor analog of the Singular Value Decomposition (SVD) for a matrix and has many data science applications including signal processing and machine learning. For the CPD, the *Alternating Least Squares (ALS)* algorithm has been used extensively. Although the ALS algorithm is simple, it is sensitive to a noise of a data tensor in the applications. In this paper, we propose a novel strategy to realize the noise suppression for the CPD. The proposed strategy is decomposed into two steps: (Step 1) denoising the given tensor and (Step 2) solving the *exact* CPD of the denoised tensor. Step 1 can be realized by solving a *structured low-rank approximation* with the *Douglas-Rachford splitting algorithm* and then Step 2 can be realized by solving the simultaneous diagonalization of a matrix tuple constructed by the denoised tensor with the DODO method. Numerical experiments show that the proposed algorithm works well even in typical cases where the ALS algorithm suffers from the so-called bottleneck/swamp effect.

ER -