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.
Riku AKEMA
Tokyo Institute of Technology
Masao YAMAGISHI
Tokyo Institute of Technology
Isao YAMADA
Tokyo Institute of Technology
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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 -