Consider a user-item dataset where every datapoint consists of information matching user U to an item I. Just as in the previous part, we can represent this with a matrix R where each row corresponds to a user and each column corresponds to an item, so that Rij = 1 means that user i likes item j and otherwise Rij = 0. We'll assume we have m users and n items, so that R is m x n. We'll now define an m x m matrix P to be a diagonal matrix whose diagonal entries are the number of items liked by person i and Q an n x n diagonal matrix is the number of users that like item i. 1. The non-normalized user similarity matrix is defined by T = RRT. Explain the interpretation of Ti,i and Tij in terms of the underlying data. 2. Define the item similarity matrix Sy to be an nxn matrix so that the i, j element is the cosine similarity of item i and item j, which corresponds to the ith and jth columns of R. Observe that SI = Q-/2 RT RQ-1/2 where Q-1/2 is Q2/2 = 1//Qr,c for all non-zero entries of Q and 0 everywhere else. 3. We can also define a user similarity matrix whose entries are the cosine similarities of the users (rows of R). Further note that we get an expression for Su in terms of R, P, and Q: P-1/2 RRT P-1/2 4. The recommendation method for user-user collaborative filtering for user u, can be described as follows: for all items s compute cos-sim(x, u) Rr,s Eitems and recommend the k items for which ru,s is the largest. Similarly, the item-item collaborative filter for user u works by evaluating: ru,s= Ru,z cos-sim (2, s) xEitems over all items s and picking the k items for which ru,s is the largest. Tu, s =