Higher Order Tensor of Variable Rank

Question

I would like to write a function, that matricizes a higher order tensor according to the following rule:

Let $\mathcal{A} \in \mathbb{C}^{I_{1} \times I_{2} \times \ldots \times I_{N}}$ be a tensor of order $N$. The matrix unfolding $\mathbf{A}_{(n)} \in \mathbb{C}^{I_{n} \times\left(I_{n+1} I_{n+2} \ldots I_{N} I_{1} I_{2} \ldots I_{n-1}\right)}$ contains the element $a_{i_{1} i_{2} \ldots i_{N}}$ at the position with row number $i_n$ and column number equal to $$\begin{array}{l}{\left(i_{n+1}-1\right) I_{n+2} I_{n+3} \ldots I_{N} I_{1} I_{2} \ldots I_{n-1}+\left(i_{n+2}-1\right) I_{n+3} I_{n+4} \ldots I_{N} I_{1} I_{2} \ldots I_{n-1}+\cdots} \\ {\quad+\left(i_{N}-1\right) I_{1} I_{2} \ldots I_{n-1}+\left(i_{1}-1\right) I_{2} I_{3} \ldots I_{n-1}+\left(i_{2}-1\right) I_{3} I_{4} \ldots I_{n-1}+\cdots+i_{n-1}}\end{array}$$

How do a write a function, that will accept a higher order tensor of variable rank $N$ as an input?

Thanks

Answer 1

This should do what you want (you might want to double-check that):

unfold[ten_, n_] := Flatten /@ Transpose[ten, RotateRight[Range@TensorRank@ten, n - 1]]

This works by first rearranging the levels of the tensor from $\{1,\dots,N\}$ to $\{n,n+1,\dots,N,1,\dots,n-1\}$ and the flattening everything but the first level.

An example:

A = Array[Subscript[a, ##] &, {3, 3, 3}];

Column@Table[MatrixForm@unfold[A, n], {n, 3}]