Volume 62, pp. 58-71, 2024.

Relaxation of the rank-1 tensor approximation using different norms

Hassan Bozorgmanesh

Abstract

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 the leading eigenvector of 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.

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Key words

rank-1 approximation, relaxation, tensors, maximum Z-eigenvalue

AMS subject classifications

15A18, 15A69

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