## 1 Introduction
However, the usual embedding-based architecture in the RS area exploits collaborative relationships, such as in [29] where they use embeddings to code user-item bipartite graphs for recommendation and representation learning. Relationships are also managed by means of low-dimensional dense embeddings learned from the sparse features in a wide and deep RS architecture [30]. A k-partite graph is used in [31] to characterize several types of information in recommendation tasks, and embeddings for different kinds of information are projected in the same latent space. A collaborative user network embedding has been proposed for social RS, where the cold start problem is addressed by combining MF and Bayesian personalized ranking [32]. A method to automatically set embedding sizes in RS [33] is based on the use of a reinforcement learning agent that adaptatively selects adequate sizes. Finally, internal embedding information is combined in [34] to obtain prediction and recommendation reliabilities. Using deep learning variational approaches, we obtain wider, more representative, and more robust latent spaces and embedding representations. Neural autoencoders are, in certain cases, reinforced with a variational stage, building Variational Auto-Encoder (VAE) models. Mainly, VAEs are particularly applied to the image processing area, e.g., reconstructing images [35], creating super-resolution images by encoding low-resolution images in a dense latent space vector [36], and reducing blurring by adding a conditional sampling mechanism [37]. This paper proposes adding variational layers to the neural model suggested to improve the latent space where the embedding samples are located. It mimics the underlying VAE operative to obtain super-resolution images, reducing blurring,and handling low-resolution samples: using VAE, the latent space is enriched, and samples are spread. Enriched embeddings are used in image processing to decode high-resolution images, unblurred images, etc., whereas we propose the use of enriched embeddings to improve the visual representation of RS information. From the explained research, this paper proposes an innovative deep learning model that incorporates two embedding layers: one for code users and the other for code items. Both embeddings will have small sizes to make it possible to draw bi- or three-dimensional graphs of user and item samples. The accuracy loss caused by the small embedding sizes (two or three neurons each embedding) will be tested in the paper. The proposed model also incorporates a variational stage, designed to spread the latent space where item and user embeddings are represented. Both user and item embeddings will be followed by their own Gaussian variational layers whose parameter values are learned in the whole neural model. The expected results are accurate low-dimensional item and user graphs, where samples are spread in a latent space area and not 'compressed' in a reduced space region, making it easier to discriminate between adjacent samples. Finally, a 'Lambda' join layer is added to the model to implement the Euclidean distance between the embeddings of the items and the embeddings of the users. This layer replaces the 'Dot' product layer of the traditional DeepMF model or the MLP stage of the NCF model. The Euclidean Lambda layer's purpose is to keep near to related user or item embeddings and to keep far from nonrelated user or item embeddings, such as humans intuitively understand distances. In short, this paper proposes a new deep learning design to obtain accurate plots of RS information. The innovative model incorporates embedding layers of small (representable) sizes, variational layers to improve the latent space and to spread samples, and a Euclidean similarity measure to place samples according to the intuitive human interpretation of distances. Experiments have been conducted using representative CF data sets to test the proposed model.