Tensor: dimension

Question

Please help me to understand the basic in tensor theory. In papers, I have found 2 expressions: 3-way tenor and 3rd tensor. What do they mean? Is it dimension in these cases? The 3-way tensor is a 3 rd dimension tensor, does it?

Answer 1

It seems the terminology 3-way tensor is equivalent to saying third order tensor and seems to be used in the machine learning community (see this article)

In tensor algebra, the order of a tensor can be identified through the number of basis vectors or the number of free indices. For example: $$\boldsymbol A=A_{ijkl}g^i\otimes g^j\otimes g^k\otimes g^l$$ would be a fourth order tensor.

Regarding your question "Is it dimension in these cases?": In physics the term dimension is rather used for the dimension of the underlying space or the domain the tensor is a function of. So there can be fourth order tensors in a three-dimensional setting.

However, I have not seen terminology like "3 rd dimension tensor" being used anywhere.