# Canonical polyadic decomposition of tensors that are common in some dimension

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)

https://arxiv.org/pdf/1607.01668.pdf

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