# Matrix factorization for recommendations vs Collaborative filtering

Some texts seem to list matrix factorization as a method for collaborative filtering, and more specifically categorize them as a "model-based approach" (e.g. here and here), while others seem to treat them differently (e.g. see here where the presenter discusses three distinct solutions, content-based, collaborative, and latent-factor-based. Similar implied distinctions are madehere or e.g. here)

This is perhaps a matter of semantics and etymology, but what are the merits for each argument? That is, from what perspective is MF a CF method ("model-based or not) or vice-versa?

• What “arguments” you have in mind?
– Tim
Commented May 29, 2020 at 18:44
• Thanks @Tim I don't actually know if there are mathematical arguments or interpretations of MF that would group MF as a model-based CF method. If there are none (perhaps because the term CF never had a sufficiently rigorous definition), the explanation may be historical context. So by "argument" here I mean what arguments would an author use to group MF under CF or not.
– Josh
Commented May 29, 2020 at 18:55

Matrix factorization is a kind of collaborative filtering,

collaborative filtering is a method of making automatic predictions (filtering) about the interests of a user by collecting preferences or taste information from many users (collaborating).

Matrix factorization is based on a model (it learns representation of the data vs just finding similar users using some kind of distance metric) that factorizes the matrix of user-item preferences $$R$$ into two latent (unobstructed) variable matrices $$P$$ and $$Q$$:

$$R \approx PQ$$

So all the descriptions are correct, they are just of different granularity.

• +1 - Thanks, although I'm not sure I would go as far as to say that all the descriptions are correct, since some of the sources above seem to list user-item matrix factorization approaches as a standalone entity outside of CF. It could be an oversight of the authors, or an unintended interpretation of their text / presentation.
– Josh
Commented May 30, 2020 at 0:31
• @Josh I didn’t watch the YouTube video, but the rest of the sources seem to be inline with what I wrote.
– Tim
Commented May 30, 2020 at 7:06