Higher order tensors in Statistics I am looking for problems in Statistics or application areas in Statistics which deal with three-way data or higher order data or 3-Tensors. Is there a regression setting where the covariates are expressed as a higher order tensor instead of a matrix (2-Tensor)?
 A: I found a reference to the following book which might be useful:
Tensor Methods in Statistics, P. McCullagh (1987)
It appears that the book has been uploaded by the author on his webpage to download for free.
A: Image recognition
If you have a black a white image (e.g. MNIST), a typical way to represent the image is an array, so a 2-tensor.
If you have a color image, you have one 2-tensor per primary color.$^{\dagger}$ Therefore, you have a 3-tensor.
So when you do a convolutional neural network to predict if a color photo is of a dog or a cat, you are inputting a 3-tensor and outputting the category, so $\mathbb{R}^{m\times n \times k} \rightarrow \{\text{dog}=0\text{ , cat}=1\}$, where $m$ is the number of rows, $n$ is the number of columns, and $k$ is the number of levels (so $k=3$ if you consider the primary colors).
If you have video, perhaps you would consider time as adding another order to the tensor to give you a 4-tensor.
$^{\dagger}$This is a typical way to do it but not the only way.
