Rating matrix

A recommendation system usually involves having a set of users U = {u₁, u₂, …, u_m} that have varying preferences on a set of items I = {i₁, i₂, …, i_n}. The number of users |U| = m is different from the number of items |I| = n in the general case. In addition, users can often express their preference by rating items on some scale. As an example, we can think of the users as being restaurant patrons in a city, and the items being the restaurants that they visit. Under this setup, the preferences of the users could be expressed as ratings on a five star scale. Of course, our generalization does not require that the items be physical items or that the users be actual people—this is simply an abstraction for the recommender system problem that is commonly used.

As an illustration, think of a dating website in which users rate other users; here, the items that are being rated are the profiles of the actual users themselves. Let's return to our example of a restaurant recommender system...