$1.$ Implement a decision tree$-$based predictor of ratings for an incomplete data sets. Use

the dimensionality reduction approach described in the chapter.

$2.$ How would you use a rule$-$based collaborative filtering system in the case where ratings

are real$-$numbers in $[1,1].$

$3.$ Design an algorithm that combines association rule methods with clustering for recommendations in order to discover localized associations in unary data. What is the

advantage of this approach over a vanilla rule$-$based method?

$4.$ The naive Bayes model discussed in this chapter predicts the ratings of each item

using the user$$s other ratings as a conditional. Design a Bayes model that uses the

$3\mathrm{.}10.$ EXERCISES $137$

item$$s other ratings as a condition. Discuss the advantages and disadvantages of each

model. Identify a case in which each approach would work better. How would you

combine the predictions of the two models?

$5.$ Suppose that a merchant had a unary matrix containing the buying behavior of various

customers. Each entry in the matrix contains information about whether or not a

customer has bought a particular item. Among the users that have not bought an

item yet, the merchant wishes to rank all the users in order of their propensity to buy

it$.$ Show how to use the Bayes model to achieve this goal.

$6.$ Use the Bayes model on Table $3\mathrm{.}1$ to determine the probability that John might buy

Bread in the future. Treat $0$s in the table as values that are actually specified for the

ratings, rather than as missing values $($except for John$$s ratings for Bread and Beef$).$

Determine the probability that he might buy Beef in the future. Is John more likely

to buy Bread or Beef in the future?

$7.$ Implement the naive Bayes model for collaborative filtering.

$8.$ Perform a straightforward rank$-2$ SVD of the matrix in Table $3\mathrm{.}2$ by treating missing

values as $0.$ Based on the use of SVD$,$ what are the predicted ratings for the missing

values of user $3?$ How does this compare with the results shown in the example of

section $3\mathrm{.}6\mathrm{.}5\mathrm{.}4,$ which uses a different initialization? How do the results compare to

those obtained using the Bayes model described in the chapter?

$9.$ Suppose you are given a matrix R which can be factorized as R $=$ UV T $,$ where the

columns of U are mutually orthogonal and the columns of V are mutually orthogonal.

Show how to factorize R into three matrices in the form Q$\backslash$Sigma P T $,$ where the columns

of P and Q are orthonormal and $\backslash$Sigma is a non$-$negative diagonal matrix.

$10.$ Implement the unconstrained matrix factorization method with stochastic gradient

descent and batch updates.

$11.$ Discuss the changes required to the alternating least$-$squares method for unconstrained

matrix factorization, when one constrains the last column of the user$-$factor matrix

to contain only $1$s$,$ and the second$-$last column of the item$-$factor matrix to contain

only $1$s$.$ This method is useful for incorporating user and item biases in unconstrained

matrix factorization.

$12.$ Discuss how you might apply the alternating least$-$squares method for designing latent

factor models with implicit feedback.