My situation:
I'm an absolute beginner regarding tensors and Ive currently working on a manuscript about a format which approximate / represent tensors of possibly high order. But before I can start with that, I need to clarify basic questions. I know they are trivial, nevertheless I hope for your help.
So first the definition of a tensor:
A tensor $ \chi \in \mathbb{C}^{{n_1} \times ...\times {n_d}} $ is a $d$-dimensional array with entries $ \chi_{i_1...i_d} \in \mathbb{C}$. Usually, $d$ is called the order of the tensor.
the first approximation is:
$ vec(\chi) = u_d \otimes u_{d-1} \otimes ... \otimes u_1 , \quad u_1 \in \mathbb {C}^{n_1}, ... , u_d \in \mathbb{C}^{n_d} $
where $vec$ stacks the entries of a tensor in reverse lexicographical order into a long column vector and $ \otimes $ denotes the standard Kronecker product.
So here is my first request:
- Can you give me an example for the first approximation?
So lets say $ \chi \in \mathbb{C}^{ 2 \times 3 } $ with
$$\left( \begin{matrix} 1 & 3 & 10 \\ 2 & 5 & 7 \\ \end{matrix} \right) $$
what exactly is $vec(\chi)$ now? Is it:
$$ \begin{pmatrix} 7 \\ 5 \\ 2 \\ 10 \\ 3 \\ 1 \end{pmatrix} $$
and what is $ u_1 \otimes u_2 $ in this special case?
- the second approximation is a more general Canonical Polyadic decomposition:
$ vec(\chi) \approx \sum_{j=1}^R u_d^{(j)} \otimes u_{d-1}^{(j)} \otimes ... \otimes u_1^{(j)} $, $ \quad u_1^{(j)} \in \mathbb {C}^{n_1}, ... , u_d^{(j)} \in \mathbb{C}^{n_d} $
so my second question: Can you explain this to me? I mean I don't know what $ R$ is. An trivial example might help me here as well.
Actually I have more questions, but I dont want to go beyond the scope.
Thank you for your help.