Latent factors are the same in both decomposed matrices? This question is in the context of recommendation systems. We can use matrix factorization techniques to decompose a user-product explicit/implicit matrix(R) into two matrices(U, P). Let's say R is a n*m matrix, U is a n*k matrix and P is a k*m matrix. U is considered as a user embedding matrix with k latent factors. P is considered as a product embedding matrix with k latent factors. Are these two sets of k latent factors considered to be the same latent factors? If so, what is the intuitive explanation for this?

The factors are the same for user and product.

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*$U_{ik}$ specifies how much user $i$ prefers factor $k$.


*$P_{kj}$ tells you about the similarity between factor $k$ and product $j$.
To get the overall preference, $j$ is decomposed into its similarity to every factor and weighted by user preference for these factors.
These weighted similarities are then just added to get the recommendation matrix $R_{ij} = \sum_{k \in \text{factors}} U_{ik} P_{kj}$
As an illustration, assume your products are movies and you consider only one factor. This factor might be the amount of action in the movie. Say, user 1 likes a lot of action $U_{11} = 1$, and user 2 doesn't $U_{21}=-1$.
A romantic comedy movie described by $P_{11} = -.1$ would receive a rather negative rating from user 1 $U_{11}P_{11} = -.1$.
User 2 on the other hand would give it a more positive one $U_{21}P_{11} = .1$.
You might want to read it up on the original post by Simon Funk where he illustrates the method rather nicely: https://sifter.org/~simon/journal/20061211.html
