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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 58-71, 2024/08/23
The best rank-1 approximation of a real $m$th-order tensor is equal to solving $m$ 2-norm optimization problems that each corresponds to a factor of the best rank-1 approximation. In this paper, these problems are relaxed by using the Frobenius and $L_1$-norms instead of the 2-norm. It is shown that the solution for the Frobenius relaxation of optimization problems is theleading eigenvectorof a positive semi-definite matrix which is closely related to higher-order singular value decomposition and the solution of the$L_1$-relaxation can be obtained efficiently by summing over all modes of the associated tensor but one. The numerical examples show that these relaxations can be used to initialize the alternating least-squares (ALS) method and they are reasonably close to the solutions obtained by the ALS method.
Keywords: rank-1 approximation, relaxation, tensors, maximum Z-eigenvalue