Collaborative Filtering Doubt on Cost Function

As seen in the above image, the cost function to learn w & b parameters has 2 summation functions. I understand that the inner summation goes over all movies for a particular user where there exists a rating. This helps learn parameters w1, w2, …, wn & b1, b2, …, bn for a single user. However, why is there another summation outside this? To find the parameters for each user don’t we just have to repeat the same process? The summation outside suggests to me that there is some connection between the weights for each user. Similarly, why does this exist for learning x1, x2, … xn?

We are optimizing parameters for all users considering all users and movies at a time, and that’s why both summations are needed. As the lab for collaborative filtering says

This is the source of the name of this approach - all the users collaborate to generate the rating set.

Yes - there are connections between users and movies. Movie rated by only 1 user would have got a different movie vector if it had been rated by 10 users instead. 2 users rated the same set of movies with similar scores should get similar user vectors.


Okay that makes sense! I was just confused about this as they had put the double summation even before introducing the collaborative filtering approach.

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That’s great! @Chandni_Kausika

@rmwkwok I see some online resources speaking about “similar users” and “similar items” in the context of collaborative filtering. Does “similar users” imply similar weight vectors(w), in the equation above? Does “similar items” mean movies with similar feature vectors (x)?

Hi @Alexander_Leon, don’t miss out b when you talk about users, then I think so.