## Orthogonal tensor decompositions.(English)Zbl 1005.15020

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

Zbl 0889.65035
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