## 2 Models and Methods  
The current deep CF state of the art includes two remarkable neural models: DeepMF (Figure 1a) and NCF (Figure 1b). As shown in Figure 1, both DeepMF (Figure 1a) and NCF (Figure 1b) models provide two embedding layers: the first codes users and the second codes items. These are the embeddings that this paper addresses. DeepMF (Figure 1a) uses only a dot product to combine user and item factors, as well as the MF machine learning method. It is simple and it provides accurate results; nevertheless, it does not catch the nonlinear complex relations existing among users and items embedding. To solve the drawback mentioned, the NCF model (Figure 1b) incorporates an MLP that non-linearly combines factors of the user and the item, returning scalar regression values (predictions). Previously, a concatenate layer joined the embedding values of the user and the item and provided a single tensor flow to the MLP. The embedding layers of the existing CF models are not designed for visual representations due to the following reasons: (1) they are vectors of excessive large sizes to be  
Figure 1: Deep matrix factorization (**a**) vs. neural collaborative filtering (**b**) recommender system models. visualized, (2) their values tend to cluster in small representation areas, and (3) The neural learning process does not consider visually understandable similarity measures (such as the Euclidean distance). To tackle the aforementioned drawbacks, we will provide three different contributions: (1) Testing the accuracy impact of reducing the embedding sizes just to two or three dimensions, (2) Expanding the embedding values representation by using the variational approach, and (3) Incorporating the Euclidean similarity measure in the deep neural model. Contribution 1: In the CF field, it is particularly useful to visually represent users and items in such a way that clients, company managers, and technical staff can understand the existing relations among users, items, and between both users and items. This leads us to code users and items using only two or three dimensions. The key question here is: is it affordable for the accuracy we will lose in the process? As we will show in the next section, the answer is yes, tested datasets show little significant accuracy decrease. Contribution 2: We borrow the variational method from the variational autoencoder field; they expand the embedding representation of samples, making it possible to improve clustering and classification, and to return a progressive morphing when needed. Figure 2 explains the variational approach, where each sample embedding (white circle) can be probabilistically located (grey circles) nearby (green circle) to its nonvariational fixed location (white circle). Variational methods are usually implemented by setting a Gaussian distribution in each embedding dimension. The defined set of parameters of the Gaussian distributions (blue and orange distributions) establishes the probabilistic area where the samples lay out. Our neural model specifically learns both the mean and variance of each Gaussian distribution. As explained, the variational approach expands the area where the sample embeddings lay out. This is particularly adequate for our embedding representation goal since it will make it easier to visually catch our attention on the existing sample relations. As an example, in Figure 3 we show the variational result (embeddings) of the proposed model applied to the MNIST dataset, where samples have been stochastically spread to make the classification of the classes easier. Figure 3 left and right graphs show, respectively, the obtained latent space and its cumulative normal distribution. The cumulative normal distribution is frequently used to support generative tasks; in this case, it can be used to generate fake embeddings, and then to obtain fake samples (MNIST numbers). In the CF area, this opens the door to implementing augmentation data and to obtaining augmented RS datasets.