Collaborative filtering algorithms try to solve the prediction problem (as described in the Chapter 1, Getting Started with Recommender Systems). In other words, we are given a matrix of i users and j items. The value in the ith row and the jth column (denoted by rij) denotes the rating given by user i to item j:
Our job is to complete this matrix. In other words, we need to predict all the cells in the matrix that we have no data for. For example, in the preceding diagram, we are asked to predict whether user E will like the music player item. To accomplish this task, some ratings are available (such as User A liking the music player and video games) whereas others are not (for instance, we do not know whether Users C and D like video games).