(Solved): Matrix Factorization Practice 1 point possible (graded) We now use collaborative filtering to solv ...

Matrix Factorization Practice 1 point possible (graded) We now use collaborative filtering to solve the movie recommender system problem. As we saw in the previous problem, we ended up with an unsatisfactory and trivial solution of $$X$$ by minimizing the objective alone: $$J(X)=\underset{a,i?D}{?}\frac{{\left(Y_{ai}?X_{ai}\right)}^2}{2}+\frac{?}{2}\underset{?a,i)}{?}{X}_{ai}^2.$$ In the collaborative filtering approach, we impose an additional constraint on $$X$$ : $$X=U{V}^T$$ for some $$n\times d$$ matrix $$U$$ and $$d\times m$$ matrix $$V^T$$. In this case, we say that $$X$$ has rank at least $$d$$. More pointedly, we say that the the rank of a matrix $$X$$ is the smallest number $$d$$ for which there exists some $$n\times d$$ matrix $$U$$ and $$m\times d$$ matrix $$V$$ with $$U{V}^T=X$$. We phrase it that way because there may be multiple pairs $$U,V$$, including pairs of a diversity of matrix shapes, that multiply to $$X$$. It builds intuition to observe that $${}^{?}X=U{V}^T$$ ' says that every row of $$X$$ is a linear combination of the rows of $$U$$ (and this linear combination's coefficients come from $$V$$ ). Suppose $$X=\left[\begin{array}{lll}3& 6& 3\\ 2& 4& 2\\ 1& 2& 1\end{array}\right]$$ then what is the minimum possible $$d$$ ? $$d=$$ You have used 0 of 3 attempts Save Intuition on the Vector Factors 1 point possible (graded) Assume we have a 3 by 2 matrix $$X$$ i.e. we have 3 users and 2 movies. Also, $$X$$ is given by $$X=\left[\begin{array}{cc}\text{ User 1’s rating on movie }1& \text{ User 1’s rating on movie 2 }\\ \text{ User 2’s rating on movie }1& \text{ User 2’s rating on movie 2 }\\ \text{ User 3’s rating on movie }1& \text{ User 3’s rating on movie 2 }\end{array}\right]=U{V}^T$$ for some $$3\times d$$ matrix $$U$$ and $$d\times 2$$ matrix $$V^T$$. Now which of the following is true about $$U$$ and $$V^T$$ ? (Choose all those apply.) The first row of $$U$$ represents information on user 1's rating tendency The first row of $$U$$ represents information on movie 1 The first column of $$V^T$$ represents information on user 1 's rating tendency The first column of $$V^T$$ represents information on movie 1