Exact probabilities for typical ranks of \(2 \times 2 \times 2\) and \(3 \times 3 \times 2\) tensors
Let \(T\) be a real \(m\times n\times p\) 3-tensor, then its CP expansion is given as: \[ T=\sum_{i=1}^{r}c_{i}u_{i}\otimes v_{i}\otimes w_{i}, (c_{i}\in \mathbb R,u_{i}\in \mathbb R^{m},v_{i}\in \mathbb R^n,w_{i}\in \mathbb R^{p}) \] where \(\otimes\) denotes the tensor product \textit{R. Harshman} [``Foundations of the PARAFAC procedure: models and conditions for an explanatory multi-modal factor analysis'', UCLA Working Pap. phonetics 16, 1--84 (1970)]. The rank of T is the minimal possible value of in the above expansion and is always well defined. The author of the paper under review uses the results of \textit{A. Edelman, E. Kostlan} and \textit{M. Shub} [J. Am. Math. Soc. 7, No. 1, 247--267 (1994; Zbl 0790.15017)] to show that the probability to be of rank 2 for a \(2\times 2\times 2\) tensor with elements from a standard normal distribution is \((\pi /4)\), and that the probability to be of rank \(3\) for a \(3\times 3\times 2\) tensor is \((1/2)\). The author also gives some information on the rank of \(n\times n\times 2\) tensors with \(n\geq 4\).
scientific article; zbMATH DE number 6125883