# 3. Tensor basics

## 3.1. Creating a tensor

A tensor is nothing more than a multi-dimensional array.

Let’s take for this example the tensor $$\tilde X$$ defined by its frontal slices:

\begin{align}\begin{aligned}\begin{split}X_1 = \left[ \begin{matrix} 0 & 2 & 4 & 6\\ 8 & 10 & 12 & 14\\ 16 & 18 & 20 & 22\\ \end{matrix} \right]\end{split}\\\text{and}\\\begin{split}X_2 = \left[ \begin{matrix} 1 & 3 & 5 & 7\\ 9 & 11 & 13 & 15\\ 17 & 19 & 21 & 23\\ \end{matrix} \right]\end{split}\end{aligned}\end{align}

In Python, this array can be expressed as a numpy array:

>>> import numpy as np
>>> import tensorly as tl
>>> X = tl.tensor(np.arange(24).reshape((3, 4, 2)))


You can view the frontal slices by fixing the last axis:

>>> X[..., 0]
array([[ 0,  2,  4,  6],
[ 8, 10, 12, 14],
[16, 18, 20, 22]])

>>> X[..., 1]
array([[ 1,  3,  5,  7],
[ 9, 11, 13, 15],
[17, 19, 21, 23]])


## 3.2. Unfolding

Also called matrization, unfolding a tensor is done by reading the element in a given way as to obtain a matrix instead of a tensor.

For a tensor of size $$(I_0, I_1, \cdots, I_N)$$, the n-mode unfolding of this tensor will be of size $$(I_n, I_0, I_1 \times \cdots \times I_{n-1} \times I_{n+1} \cdots \times I_N)$$.

Important

In tensorly we use an unfolding different from the classical one as defined in 1 for better performance.

Given a tensor $$\tilde X \in \mathbb{R}^{I_0, I_1 \times I_2 \times \cdots \times I_N}$$, the mode-n unfolding of $$\tilde X$$ is a matrix $$\mathbf{X}_{[n]} \in \mathbb{R}^{I_n, I_M}$$, with $$M = \prod_{\substack{k=0,\\k \neq n}}^N I_k$$ and is defined by the mapping from element $$(i_0, i_1, \cdots, i_N)$$ to $$(i_n, j)$$, with

$\begin{split}j = \sum_{\substack{k=0,\\k \neq n}}^N i_k \times \prod_{\substack{m=k+1,\\ k \neq n}}^N I_m.\end{split}$

Warning

Traditionally, mode-1 unfolding denotes the unfolding along the first dimension. However, to be consistent with the Python indexing that always starts at zero, in tensorly, unfolding also starts at zero!

Therefore unfold(tensor, 0) will unfold said tensor along its first dimension!

For instance, using the $$\tilde X$$ previously defined, the 0-mode unfolding of $$\tilde X$$:

$\begin{split}\tilde X_{[0]} = \left[ \begin{matrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15\\ 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23\\ \end{matrix} \right]\end{split}$

The 1-mode unfolding is given by:

$\begin{split}\tilde X_{[1]} = \left[ \begin{matrix} 0 & 1 & 8 & 9 & 16 & 17\\ 2 & 3 & 10 & 11 & 18 & 19\\ 4 & 5 & 12 & 13 & 20 & 21\\ 6 & 7 & 14 & 15 & 22 & 23\\ \end{matrix} \right]\end{split}$

Finally, the 2-mode unfolding is the unfolding along the last axis:

$\begin{split}\tilde X_{[2]} = \left[ \begin{matrix} 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22\\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21 & 23\\ \end{matrix} \right]\end{split}$

In tensorly:

>>> from tensorly import unfold
>>> unfold(X, 0) # mode-1 unfolding
array([[ 0,  1,  2,  3,  4,  5,  6,  7],
[ 8,  9, 10, 11, 12, 13, 14, 15],
[16, 17, 18, 19, 20, 21, 22, 23]])

>>> unfold(X, 1) # mode-2 unfolding
array([[ 0,  1,  8,  9, 16, 17],
[ 2,  3, 10, 11, 18, 19],
[ 4,  5, 12, 13, 20, 21],
[ 6,  7, 14, 15, 22, 23]])

>>> unfold(X, 2) # mode-3 unfolding
array([[ 0,  2,  4,  6,  8, 10, 12, 14, 16, 18, 20, 22],
[ 1,  3,  5,  7,  9, 11, 13, 15, 17, 19, 21, 23]])


## 3.3. Folding

You can fold an unfolded tensor back from matrix to full tensor using the tensorly.base.fold function.

>>> from tensorly import fold
>>> unfolding = unfold(X, 1)
>>> original_shape = X.shape
>>> fold(unfolding, 1, original_shape)
array([[[ 0,  1],
[ 2,  3],
[ 4,  5],
[ 6,  7]],

[[ 8,  9],
[10, 11],
[12, 13],
[14, 15]],

[[16, 17],
[18, 19],
[20, 21],
[22, 23]]])


## 3.4. References

1

T.G.Kolda and B.W.Bader, “Tensor Decompositions and Applications”, SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.