In recommender system module, low-rank matrix factorization was used to execute latent factor modeling of movie ratings data. Specifically, we calculated matrices U and I to solve the following optimization problem (if all ratings were given): U.V U.V min f(U, V) = min ||R - VUll? = min { > >(m - umu;)= U.V m=1 i= 1 The best U and I/' were calculated iteratively by improving on current estimates: up = u; + 2a(mi - Umu! ) . Um Unew = Um + 2a(1mi - Umll; ) . u,, where a is the step-size that is to be chosen by the user. (We won't discuss the role in this class, but treat it as an arbitrary, but given, parameter) We can make calculating the updates more efficient by calculating them with matrix operations. For example, instead of calculating each deviation Ymi = I'mi - Umu; separately for all m = 1, 2, ..., M and i = 1, 2, ..., I, matrix I of all deviations can be computed together using matrix operation _(verify for yourself)_: T = R - VUT Similarly, updating U and I can be combined into matrix calculations which makes the optimization procedure more efficient. First, note that updates for w/ , i = 1, 2, ..., I can be rewritten as ynew = u1 + 2ayml . Um 42 * = u2 + 207m2 . Um new = up + 207ml . Um. Stacking all I equations into a matrix form, [new = U + 2alm_Um, where I'm- is the m-th row of I' (use the notation I'_, for the i-th column). Note that there are M such update equations (one for each m = 1, 2, ..., M) that can also be combined into one matrix update equation involving matrices U, I, I' and scalars. As stated earlier, since a is assumed to be an arbitrary step-size parameter, we can replace a/M with a. Complete the following update equations: (new = U + 2a[some function of [some function of V] new = V + 2a[some function of [some function of U]