def gradient$($df$,$ Y$,$ emb$\_$user, emb$\_$movie$)$:

$"""$Computes the gradient of the mean squared error with respect to user and movie embeddings.

Arguments:

df: Dataframe with user ratings.

Y: Sparse representation of observed ratings.

emb$\_$user: User embeddings.

emb$\_$movie: Movie embeddings.

Returns:

d$\_$emb$\_$user: Gradient of the MSE with respect to user embeddings.

d$\_$emb$\_$movie: Gradient of the MSE with respect to movie embeddings.

$"""$