I was studying about "Tensor Decompositions and it's Applications" from a small handout, i.e Tensor Decompositions and Applications https://www.kolda.net/publication/TensorReview.pdf.
This topic is relatively new to me. So, I need to start from basic stuffs. In addition, I am self studying this text. Naturally, I am having some questions that needs to be addressed.
I dont understand the meaning of "Fibres".
They say that,
Fibers are the higher order analogue of matrix rows and columns. A fiber is defined by fixing every index but one. A matrix column is a mode-$1$ fiber and a matrix row is a mode-$2$ fiber. Third-order tensors have column, row, and tube fibers, denoted by $x_{:jk}, x_{i:k},$ and $x_{ij:}$ , respectively; see Figure $2.1.$ When extracted from the tensor, fibers are always assumed to be oriented as column vectors.
What I understood from the line,
Fibers are the higher order analogue of matrix rows and columns. A fiber is defined by fixing every index but one.
is that, the rows and columns of each matrix is assumed to be a long rod or a stick. Each such long rod, is obtained by fixing all the indexes excluding one particular index. Now as that one unfixed index varies, we get similar rods that are oriented in the same direction.
Next, they remarked that:
A matrix column is a mode-$1$ fiber and a matrix row is a mode-$2$ fiber.
What I understood from here, was that the matrix rows are called mode-1 fiber, etc and these are just some names given.
My problem is with this line:
Third-order tensors have column, row, and tube fibers, denoted by $x_{:jk}, x_{i:k},$ and $x_{ij:}$ , respectively; see Figure $2.1.$
I don't get why Figure 2.1 (a) and (b) looks like that. I feel that the Figure 2.1 (a) and (b) should be interchanged.
I need some help to understand how did they arrive at such structures and orientations in the above Figure 2.1.