The prediction problem

In this version of the problem, we are given a matrix of m users and n items. Each row of the matrix represents a user and each column represents an item. The value of the cell in the i^th row and the j^th column denotes the rating given by user i to item j. This value is usually denoted as r_ij. 

For instance, consider the matrix in the following screenshot:

This matrix has seven users rating six items. Therefore, m = 7 and n = 6. User 1 has given the item 1 a rating of 4. Therefore, r₁₁ = 4.

Let us now consider a more concrete example. Imagine you are Netflix and you have a repository of 20,000 movies and 5,000 users. You have a system in place that records every rating that each user gives to a particular movie. In other words, you have the rating matrix (of shape 5,000 × 20,000) with you.

However, all your users will have seen only a fraction of the movies you have available on your site; therefore, the matrix you have is sparse. In other words, most of the entries in your matrix are empty, as most users have not rated most of your movies.

The prediction problem, therefore, aims to predict these missing values using all the information it has at its disposal (the ratings recorded, data on movies, data on users, and so on). If it is able to predict the missing values accurately, it will be able to give great recommendations. For example, if user i has not used item j, but our system predicts a very high rating (denoted by _ij), it is highly likely that i will love j should they discover it through the system.