Exact probabilities for typical ranks of \(2 \times 2 \times 2\) and \(3 \times 3 \times 2\) tensors
type: Item
instance of: scholarly article
title: Exact probabilities for typical ranks of \(2 \times 2 \times 2\) and \(3 \times 3 \times 2\) tensors
zbMATH Open document ID: 1266.15037
author: Göran Bergqvist
published in: Linear Algebra and its Applications
publication date: 2013-01-16T00:00:00Z
review text: 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 \(n\geq 4\).
reviewed by: Manouchehr Misaghian
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15B52
zbMATH DE Number: 6125883
zbMATH Keywords: tensors
zbMATH Keywords: multi-way arrays
zbMATH Keywords: typical rank
zbMATH Keywords: random matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2011.02.041
OpenAlex ID: W2072139290
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DOI: 10.1016/J.LAA.2011.02.041
description: scientific article; zbMATH DE number 6125883