The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. ex. Some numerals are expressed as "XNUMX".
Copyrights notice
The original paper is in English. Non-English content has been machine-translated and may contain typographical errors or mistranslations. Copyrights notice
Diagonalisasi Serentak Anggaran (ASD) ialah masalah untuk mencari penjelmaan persamaan sepunya yang lebih kurang menyerongkan tuple matriks segi empat sama tertentu. Banyak masalah sains data telah dikurangkan kepada ASD melalui pemodelan yang bijak. Untuk ASD, apa yang dipanggil seperti Jacobi kaedah telah digunakan secara meluas. Walau bagaimanapun, kaedah tersebut tidak mempunyai jaminan untuk menyekat magnitud masukan luar pepenjuru bagi tuple yang diubah walaupun jika tupel yang diberikan mempunyai diagonalizer sepunya yang tepat, iaitu, tupel yang diberikan adalah serentak boleh diagonal. Dalam makalah ini, untuk mewujudkan strategi alternatif yang berkuasa untuk ASD, kami membentangkan strategi dua langkah novel, yang dipanggil Anggaran-Kemudian-Diagonal-Serentak (ATDS) algoritma. Algoritma ATDS menguraikan ASD menjadi (Langkah 1) mencari tuple boleh diagonal serentak berhampiran yang diberikan; dan (Langkah 2) mencari penjelmaan persamaan biasa yang menyerong tepat tuple yang diperolehi dalam Langkah 1. Pendekatan yang dicadangkan untuk Langkah 1 direalisasikan dengan menyelesaikan satu Anggaran Peringkat Rendah Berstruktur (SLRA) bersama Algoritma Cadzow. Dalam Langkah 2, dengan mengeksploitasi idea dalam pembuktian membina mengenai syarat untuk kebolehpenjurongan serentak yang tepat, kami memperoleh penjepit sepunya yang tepat bagi tuple yang diperolehi dalam Langkah 1 sebagai penyelesaian untuk ASD asal. Tidak seperti kaedah seperti Jacobi, algoritma ATDS mempunyai jaminan untuk mencari diagonalizer sepunya yang tepat jika tuple yang diberikan kebetulan boleh diagonal secara serentak. Eksperimen berangka menunjukkan bahawa algoritma ATDS mencapai prestasi yang lebih baik daripada kaedah seperti Jacobi.
Riku AKEMA
Tokyo Institute of Technology
Masao YAMAGISHI
Tokyo Institute of Technology
Isao YAMADA
Tokyo Institute of Technology
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Salinan
Riku AKEMA, Masao YAMAGISHI, Isao YAMADA, "Approximate Simultaneous Diagonalization of Matrices via Structured Low-Rank Approximation" in IEICE TRANSACTIONS on Fundamentals,
vol. E104-A, no. 4, pp. 680-690, April 2021, doi: 10.1587/transfun.2020EAP1062.
Abstract: Approximate Simultaneous Diagonalization (ASD) is a problem to find a common similarity transformation which approximately diagonalizes a given square-matrix tuple. Many data science problems have been reduced into ASD through ingenious modelling. For ASD, the so-called Jacobi-like methods have been extensively used. However, the methods have no guarantee to suppress the magnitude of off-diagonal entries of the transformed tuple even if the given tuple has an exact common diagonalizer, i.e., the given tuple is simultaneously diagonalizable. In this paper, to establish an alternative powerful strategy for ASD, we present a novel two-step strategy, called Approximate-Then-Diagonalize-Simultaneously (ATDS) algorithm. The ATDS algorithm decomposes ASD into (Step 1) finding a simultaneously diagonalizable tuple near the given one; and (Step 2) finding a common similarity transformation which diagonalizes exactly the tuple obtained in Step 1. The proposed approach to Step 1 is realized by solving a Structured Low-Rank Approximation (SLRA) with Cadzow's algorithm. In Step 2, by exploiting the idea in the constructive proof regarding the conditions for the exact simultaneous diagonalizability, we obtain an exact common diagonalizer of the obtained tuple in Step 1 as a solution for the original ASD. Unlike the Jacobi-like methods, the ATDS algorithm has a guarantee to find an exact common diagonalizer if the given tuple happens to be simultaneously diagonalizable. Numerical experiments show that the ATDS algorithm achieves better performance than the Jacobi-like methods.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2020EAP1062/_p
Salinan
@ARTICLE{e104-a_4_680,
author={Riku AKEMA, Masao YAMAGISHI, Isao YAMADA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Approximate Simultaneous Diagonalization of Matrices via Structured Low-Rank Approximation},
year={2021},
volume={E104-A},
number={4},
pages={680-690},
abstract={Approximate Simultaneous Diagonalization (ASD) is a problem to find a common similarity transformation which approximately diagonalizes a given square-matrix tuple. Many data science problems have been reduced into ASD through ingenious modelling. For ASD, the so-called Jacobi-like methods have been extensively used. However, the methods have no guarantee to suppress the magnitude of off-diagonal entries of the transformed tuple even if the given tuple has an exact common diagonalizer, i.e., the given tuple is simultaneously diagonalizable. In this paper, to establish an alternative powerful strategy for ASD, we present a novel two-step strategy, called Approximate-Then-Diagonalize-Simultaneously (ATDS) algorithm. The ATDS algorithm decomposes ASD into (Step 1) finding a simultaneously diagonalizable tuple near the given one; and (Step 2) finding a common similarity transformation which diagonalizes exactly the tuple obtained in Step 1. The proposed approach to Step 1 is realized by solving a Structured Low-Rank Approximation (SLRA) with Cadzow's algorithm. In Step 2, by exploiting the idea in the constructive proof regarding the conditions for the exact simultaneous diagonalizability, we obtain an exact common diagonalizer of the obtained tuple in Step 1 as a solution for the original ASD. Unlike the Jacobi-like methods, the ATDS algorithm has a guarantee to find an exact common diagonalizer if the given tuple happens to be simultaneously diagonalizable. Numerical experiments show that the ATDS algorithm achieves better performance than the Jacobi-like methods.},
keywords={},
doi={10.1587/transfun.2020EAP1062},
ISSN={1745-1337},
month={April},}
Salinan
TY - JOUR
TI - Approximate Simultaneous Diagonalization of Matrices via Structured Low-Rank Approximation
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 680
EP - 690
AU - Riku AKEMA
AU - Masao YAMAGISHI
AU - Isao YAMADA
PY - 2021
DO - 10.1587/transfun.2020EAP1062
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E104-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 2021
AB - Approximate Simultaneous Diagonalization (ASD) is a problem to find a common similarity transformation which approximately diagonalizes a given square-matrix tuple. Many data science problems have been reduced into ASD through ingenious modelling. For ASD, the so-called Jacobi-like methods have been extensively used. However, the methods have no guarantee to suppress the magnitude of off-diagonal entries of the transformed tuple even if the given tuple has an exact common diagonalizer, i.e., the given tuple is simultaneously diagonalizable. In this paper, to establish an alternative powerful strategy for ASD, we present a novel two-step strategy, called Approximate-Then-Diagonalize-Simultaneously (ATDS) algorithm. The ATDS algorithm decomposes ASD into (Step 1) finding a simultaneously diagonalizable tuple near the given one; and (Step 2) finding a common similarity transformation which diagonalizes exactly the tuple obtained in Step 1. The proposed approach to Step 1 is realized by solving a Structured Low-Rank Approximation (SLRA) with Cadzow's algorithm. In Step 2, by exploiting the idea in the constructive proof regarding the conditions for the exact simultaneous diagonalizability, we obtain an exact common diagonalizer of the obtained tuple in Step 1 as a solution for the original ASD. Unlike the Jacobi-like methods, the ATDS algorithm has a guarantee to find an exact common diagonalizer if the given tuple happens to be simultaneously diagonalizable. Numerical experiments show that the ATDS algorithm achieves better performance than the Jacobi-like methods.
ER -