Amazon's item-to-item collaborative filtering recommender

The recommender algorithm that Amazon adopted early on was an improvement of the Recommender1 implemented previously. The main difference is in step 5, where now we have two more substeps, which pick the n₁ most similar items and sort them by popularity:

Recommender algorithm 2 is as follows:

1. Initialize the utility matrix (u_ij) with m rows and n columns, where m is the number of users and n is the number of items.

2. For each pair (i, j) in the input list, set u_ij = 1.

3. Initialize the similarity matrix (s_jk) with n rows and n columns.

4. For each j = 1…n and each k = 1…n, set s_jk = s(u, v), the cosine similarity of the j^th column u and the k^th column v of the utility matrix.

5. For a given user-purchase pair (i, j) (that is, u_ij = 1):

   • Find the set S of items not bought by user i

   • Sort the items in S according to how similar they are to item j

   • Let S' be the top n₁ elements of S

   • Sort the items in S' by popularity

6. Recommend the top n₂ items in S'.

This...