In order to create the dense model, we will substitute the Dot layer with a series of Dense layers. Dense layers are classic neurons, where each neuron gets, as input, all the outputs from the previous layer. In our case, as we have two Embedding layers, we must first concatenate them using the Concatenate layer, and then feed them to the first Dense layer. These two layers are also included in the keras.layers package. Thus, our model definition will now look like this:

# Movie part. Input accepts the index as input
# and passes it to the Embedding layer. Finally,
# Flatten transforms Embedding's output to a
# one-dimensional tensor.
movie_in = Input(shape=[1], name="Movie")
mov_embed = Embedding(n_movies, fts, name="Movie_Embed")(movie_in)
flat_movie = Flatten(name="FlattenM")(mov_embed)

# Repeat for the user.
user_in = Input(shape=[1], name="User")
user_inuser_embed = Embedding(n_users, fts, name="User_Embed")(user_in)
flat_user = Flatten(name="FlattenU")(user_inuser_embed)

# Concatenate the Embedding layers and feed them 
# to the Dense part of the network
concat = Concatenate()([flat_movie, flat_user])
dense_1 = Dense(128)(concat)
dense_2 = Dense(32)(dense_1)
out = Dense(1)(dense_2)

# Create and compile the model
model = Model([user_in, movie_in], out)
model.compile('adam', 'mean_squared_error')

By adding these three Dense layers, we have increased the number of trainable parameters from almost 52,000 to almost 57,200 (an increase of 10%). Furthermore, each step now needs almost 210 microseconds, which increased from 144 us (a 45% increase), as is evident from the training progression and summary, as depicted in the following diagrams:

Nonetheless, the model now achieves an MSE 0.77 , which is 60% of the original dot-product model. Thus, as this model outperforms the previous model, we will utilize this architecture for our stacking ensemble. Moreover, as each network has a higher degree of freedom, it has a higher probability of diversifying from other base learners.