Answer Happy

Matrix Factorization Problem 3. (15 points) Recall the following objective for extracting latent features from a partially observed rating matrix via matrix factorization (MF) for making recommendations, discussed in the class: min U V (vw.v) = [ (Ps - 10)+o] | 6.3 + . (1) ( where •n: number of users • m: number of items •RERXminput partially observed rating matrix •125 [n]X[m): index of observed entries in rating matrix, where [n] denotes the sequence of numbers {1,2,...,n}. • k: number of latent features 2 .U e Rnxk: the (unknown) matrix of latent feature vectors for n users (the ith row u; eRk is the latent features for ith user) .VERmxk the (unknown) matrix of latent feature vectors for m items (the jth row v; e Rk is the latent features for jth movie) Please do the followings: 1. In solving Equation (1) with iterative Alternating Minimization algorithm (fixing (4) and taking gradient step for U) and vice verse), discuss what happens if U) and V) are initialized to zero? 2. Discuss why when there is no regularization in basic MF formulated in Equation (1), i.e., a = B = 0, each user must have rated at least k movies, and each movie must have been rated by at least k users. 2 3. Computing the closed form solution in part (2) could be computational burden for large number of users or movies. A remedy for this would be using iterative optimization algorithms such as Stochastic Gradient Descent (SGD). Assume we run SGD for T iter- ations, where at each iteration t =1,2,..., T we sample a rating () € uniformly at random and update the latent features for user it and movie jt. Derive the updating rules for and wat tth iteration of SGD. Show the detailed steps and write the pseudo code clearly.