Matrix Factorization Problem $3.[30$ 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: $[$s$0,0)=$ min $($U$,$V$)=$ U V $($i $($ijen I $($ris $$ u$!0$;$)2+$ a $||2$:$13+$ B$$le; $(1)$ i$=1$ j$=1$ where $.$ n: number of users $$ m: number of items $$ riji $($i$,$ j$)-$th element in RE RX input partially observed rating matrix $\mathrm{.}2[$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 $$ UeRnxk: the $($unknown$)$ matrix of latent feature vectors for n users $($the ith row Ui E Rk is the latent features for ith user$)2$ 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 V $($t$)$ and taking gradient step for U$($t$)$ and vice versa$),$ discuss what happens if U$0$ and V$0)$ 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 $($Hint: consider the closed form solution for u; and v; in notes and argue why these conditions are necessary without regularization$).3.$ 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$)$ where you sample movies in updating the latent features for users and sample users in updating the latent features of movies. Derive the updating rules for w$"$ and v$.$ at tth iteration of SGD$.$ Show the detailed steps.