Let \(V:=\mathbb{R}^{n_1}\otimes\mathbb{R}^{n_2}\otimes \dots \otimes\mathbb{R}^{n_p}\) and let \(X\) be the set of rank-\(1\) tensors in \(V\). Define \(d_{v}(x)\) to be the squared Euclidean distance from \(v\in V\) to \(x\in X\). The authors are interested in analyzing the pairs \((v,x)\) such that \(d_{v}(x)\) is a critical point of \(d_{v}\); these are called the critical rank-\(1\) approximations to \(v\). In the case \(p=2\), we can identify \(V\;\)with the set of all \(n_1\times n_2\) matrices \(v\) and the critical rank-\(1\) approximations are \(x=\sigma x_{1}x_{2}^{T}\) where \(\sigma\) is a singular value of \(v\) and \(x_{1}\),\(x_{2}\) are corresponding left and right eigenvectors. In particular, \(d_{v}(x)\) is minimized when \(\sigma\) is the largest singular value of \(v\). For \(p>2\), the situation is more complicated and no simple description of the (real) critical rank-\(1\) approximations is known.
In the present paper, the authors prove a formula for the expected number \(E_{V}\) of critical points of \(d_{v}\) on \(X\) as \(v\) runs over a multivariate Gaussian distribution with values in \(V\). The formula for \(E_{V}\) involves a multidimensional integral and is too complicated to give here, but it enables the authors to use Monte Carlo methods to estimate \(E_{V}\) for some smaller dimensional examples.