zbMATH — the first resource for mathematics

On the best rank-1 and rank-\((R_1,R_2,. . .,R_N)\) approximation of higher-order tensors. (English) Zbl 0958.15026
The authors discuss a multilinear generalization of the best rank-\(R\) approximation problem for matrices: the approximation of a given higher-order tensor, in an optimal least-squares sense, by a tensor that has prespecified column rank value, row rank value, etc. For matrices, the solution is conceptually obtained by truncation of the singular value decomposition. They discuss higher-order generations of the power method and the orthogonal iteration method.
In Section 3 (Higher-order power iteration), they investigate how a given tensor can be approximated. In Section 4 (Higher-order orthogonal iteration), they generalize the best rank-1 approximation problem of Section 3. The derivation of the computational procedure follows a similar scheme.

15A69 Multilinear algebra, tensor calculus
15A18 Eigenvalues, singular values, and eigenvectors
65F20 Numerical solutions to overdetermined systems, pseudoinverses
PDF BibTeX Cite
Full Text: DOI