Consider a website in which users give ratings of "thumbs up" or "thumbs down" to items. We can encode "thumbs up" by user i on item j as yij=1 and a "thumbs down" by user i on item j as yij =0. We can use a version the matrix factorization model called Logistic MF. In this version we use a sigmoid after the dot product. The model is y^ij=(uivj) where (x)=(1+ex)1 is the sigmoid function and yij=1 is either 0 or 1 (or empty). You have nu users, nm items and K is the embedding size. y^ij is interpreted as a probability and the loss function is the binary cross entropy loss: N1[(i,j):rij=1yijlog(y^ij)+(1yij)log(1y^ij)], where N is the number of ratings with yij either 0 or 1 . Here rij=1 if there is a rating (0 or 1) of item j by user i and 0 otherwise. 1. How many parameters do you have in this model? 2. Given the following test set with predictions, compute the test loss: 3. Suppose you run your optimization and get the following U and V matrices. U=0.21.62.60.92.01.90.82.41.00.60.70.40.50.90.12.01.50.20.32.30.9,V=0.03.60.02.53.50.04.00.11.00.10.90.2 Write a numeric expression for the value of y^13 in terms of the sigmoid function. Assume matrix indices start at 1 . 4. What is the value of K ? 5. Think about how the gradient calculations for this model differ from the previous model. Go ahead and write them if you want more practice but no need to submit this part