Collaborative Filtering¶
It is an approach that uses 2 sources of data. First is the between user similarity and the second is the between item similarity.
Lets image that we want to recommend M items to a set of N users, where a user if owns an item it will give it an rating \(R_{ij} = \text{rating of ith user of the jth item}\), or \(R_{ij} = 0\) if the user has not rated the item. This gives an rise to a an \(N \times M\) matrix of ratings. If we have a large number of items and users this matrix will be sparse. Now we assume that each user and each user has an latent vector of dimennsion \(D\). For users this vector describes his preferences, and for items this vector discribes its profile.
Now our goal si to find an \(U = N \times D\) matrix that describes the preferences of each user and an \(V = M \times D\) matrix that describes each item in terms of its latent variable.
Now if we take the product of \(U \cdot V\) now we get an approximation how a user would reate an item he has not seen.