I have a tensor three dimension tensor $$H$$ of size 4x8x128. I divide this tensor into two sub tensors (ill be using matlab language to demonstrate)
$$H_1 = H(:,:,1:64)$$ $$H_2= H(:,:, 65:128)$$
Then I perform low rank ( example rank 3) canonical polyadic decomposition (CPD) of both $H_1$ and $H_2$ to obtain the factor matrices
$$ A_1, B_1, C_1$$
$$ A_2,B_2,C_2$$ Note that $A_1$ is if size 4x3 $B_1$ of size 8x3 and $C_1$ of size 64x3. The CPD is solved using alternate least square. I linked paper that shows how one can do that (page 5)
My question is does it make sense that somehow, the factor matrices $A_1$ and $A_2$ should be related? and what about $B_1$ and $B_2$? As you see the only difference between my two tensors $H_1$ and $H_2$ is the third dimension, does that mean only the third factor matrix should be different? If that is not that case, is there a way for me to actually show the relationship between the factor matrices? how can I compare the factor matrices? in other words, Id like to know what the relationship between the factor matrices are, and currently have no idea in mind on how I can do that...
Is there a correlation metric one can think of? Thank you