How is the input given to the NeuMF architecture?

I was going through this neural recommendation paper (Fig. 2). I want to implement it from scratch in Tensorflow. The thing I don't understand is how is the input given to this architecture. Can someone explain with a small example?

If I am right the embedding latent factor has to be $M\times K$ and $N\times K$ after passing through the embedding layer?

If I have $Users \times Items$ rating matrix-

[[2, NaN, 4],
 [3, 1, NaN],
 [4, NaN, 5]]

They have used the interaction/implicit matrix-

[[1, 0, 1],
 [1, 1, 0],
 [1, 0, 1]]

2. Why they have used Log Loss? Is this binary classification? If yes, aren't we supposed to find mean squared error to find the missing rating?