Orthogonal tensor decompositions (Q2784344)

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 \textit{D. Leibovici} and \textit{R. Sabatier} [Linear Algebra Appl. 269, 307-329 (1998; Zbl 0889.65035)]. NEWLINENEWLINENEWLINEThe 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.