I am reading Matrix Methods in Data Mining and Pattern Recognition by Lars Eldén. in chapter 8 there is a method for classifying handwriting digits using higher-order SVD (HOSVD). Here is a summary.
Each image is a $64 \times 64$ matrix, we put them together and create a 3D tensor. We construct such tensor for each of the classes (0,1,2,..., 9).Then for a new digit(a $64 \times 64$ matrix), one can predict its class using:
$min_z|| Z - \sum_{j} z_j A_j||\\ \mathcal{A} = \sum_{1}^k A_i \times_3 u_i^{(3)}\\ A_j = \mathcal{S(:,:,j)} \times_1U_1 \times_2U_2 $
Here, the classification is not important for me. My problem is that I don't know how to solve the above minimization when $Z$ is a 4D tensor. Here is a solution to above minimization:
In my problem I have a 5D tensor for each class, so for a new input $A_j = \mathcal{S(:,:,j)} \times_1U_1 \times_2U_2 \times_3U_3 \times_4U_4 $ and $Z$ would be 4D tensors and I dont know how to solve least squares for them.