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论文类型：期刊论文

发表时间：2015-01-01

发表刊物：SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS

收录刊物：SCIE、EI、Scopus

卷号：36

期号：1

页面范围：1-19

ISSN号：0895-4798

关键字：orthogonal tensor decomposition; low rank approximation; alternating least squares; high-order power method; polar decomposition; global convergence; Zariski topology

摘要：With the notable exceptions of two cases-that tensors of order 2, namely, matrices, always have best approximations of arbitrary low ranks and that tensors of any order always have the best rank-1 approximation, it is known that high-order tensors may fail to have best low rank approximations. When the condition of orthogonality is imposed, even under the modest assumption of semiorthogonality where only one set of components in the decomposed rank-1 tensors is required to be mutually perpendicular, the situation is changed completely-orthogonal low rank approximations always exist. The purpose of this paper is to discuss the best low rank approximation subject to semiorthogonality. The conventional high-order power method is modified to address the desirable orthogonality via the polar decomposition. Algebraic geometry technique is employed to show that for almost all tensors the orthogonal alternating least squares method converges globally.