Orthogonal tensor decompositions. (English) Zbl 1005.15020

Kolda, Tamara G.

SIAM J. Matrix Anal. Appl. 23, No. 1, 243-255 (2001).

The singular value decomposition of a real \(m\times n\) matrix can be reformulated as an orthogonal decomposition in the tensor product \(\mathbb{R}^m \otimes \mathbb{R}^n\). The present paper is concerned with possible generalizations to multiple tensor products \(\mathbb{R}^{m_1} \otimes\cdots \otimes \mathbb{R}^{m_k}\), a prime consideration being whether an analogue of the Eckart-Young approximation theorem holds. Several definitions of orthogonality are put forward and their differences carefully illustrated on examples. With the Eckart-Young theorem in mind, the author describes a “greedy tensor decomposition” close in spirit to the SVD-\(k\) process of D. Leibovici and R. Sabatier [Linear Algebra Appl. 269, 307-329 (1998; Zbl 0889.65035)].
The author claims to present, in section 5, a counterexample to Leibovici and Sabatier’s extension of the Eckart-Young Theorem. However, several points remain unclear to the reviewer. First, the author’s Example 5.1 and Leibovici and Sabatier’s Theorem 2 refer to different definitions of orthogonality. Second, further argument seems to be required in Example 5.1 to show that \(A_1\) and \(A_2\) are the best rank 1 and 2 approximations as claimed.

Reviewer: G.E.Wall (Sydney)

MSC:

15A69	Multilinear algebra, tensor calculus
62H25	Factor analysis and principal components; correspondence analysis

Keywords:

tensor decomposition; singular value decomposition; principal components analysis; multidimensional arrays; multiple tensor products; Eckart-Young theorem

Citations:

Zbl 0889.65035