## 3 Experiments and Results  
To run the designed experiments, we have chosen a set of open and representative CF databases. Table 1 shows the main parameter values of the selected datasets: MovieLens 100K [38], MovieLens1M [38], and a subset (Netflix*) of the Netflix database [39]. Please note the high number of Netflix* users compared to the MovieLens datasets. The chosen datasets have a similar structure, where their kernel is the CF information of ratings stored in files containing tuples: <user_id, item_id, rating>. Basically, they differ from each other in their sizes: number of users, items, and ratings. Additionally, the combination of the previous values determines the sparsity of the CF data. Please note that MovieLens 100K and MovieLens 1M not only differ in their number of ratings, but also in the number of users and items, and consequently in their sparsity (Table 1). Since the MovieLens 1M version is richer than the MovieLens 100K, its accuracy will be also better, as we will see in Table 2.  
From the aforementioned contributions, we will provide three different experiments to substantiate the proposed neural model: (1) CF quality impact by setting different embedding sizes, (2) numerical improvement of the proposed model versus the DeepMF baseline, (3) visual improvement of the proposed model versus the DeepMF baseline. Experiment 1: This experiment tests the 'Contribution 1' assessment stated in the preceding section. As explained in the preceding section, it is necessary to test the RS accuracy when a bottleneck is set to the embedding layers. Since we need to visually represent embedding samples, we use embedding sizes: 2 (two-dimensional representation) or 3 (three-dimensional representation), whereas the usual implementation sizes range from 5 to 10. Our first experiment tests the accuracy loss when small embedding sizes are set. For each tested dataset (Table 1), we obtain the Mean Absolute Error (MAE) by setting embedding sizes = {2, 3, 5, 10}. Table 2 shows the MAE results, as well as the achieved accuracy percentage comparing the embedding sizes 2 and 10. As can be seen, very little accuracy is lost setting visualizable embedding sizes (2 and 3) compared to the usual sizes (5 to 10). Notably, only 2% of accuracy is lost in the worst-case scenario. The results in Table 2 open the door to visually represent the sample embeddings of items and users, knowing that the embedding values are meaningful to provide accurate CF predictions. Experiment 2: This experiment numerically shows the improvement obtained by combining the three contributions stated in the preceding section. Once we have validated the adequacy of using visualizable embedding sizes, it is time to test the obtained improvement using the proposed approach. We will test visual improvement using the standard \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **\#Users** & **\#Items** & **\#Ratings** & **Scores** & **Sparsity** \\ \hline Movielens 100K & 943 & 1682 & 99,831 & 1 to 5 & 93.71 \\ MovieLens 1M & 6040 & 3706 & 911,031 & 1 to 5 & 95.94 \\ Netflix* & 23,012 & 1750 & 535,421 & 1 to 5 & 98.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the main parameters of the tested datasets. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Dataset\(\backslash\)Embedding Size**} & \multicolumn{3}{c}{**Mean Absolute Error (MAE)**} & **Achieved** \\ \cline{2-5}  & **2** & **3** & **5** & **10** & **Accuracy** \\ \hline Movielens 100K & 0.7355 & 0.7368 & 0.7297 & 0.7213 & 98.04\% \\ MovieLens 1M & 0.6927 & 0.6875 & 0.6839 & 0.6801 & 98.15\% \\ Netflix* & 0.7260 & 0.7250 & 0.7248 & 0.7243 & 99.76\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean absolute error results using the proposed variational Euclidean method (embedding sizes = 2, 3, 5, 10 and the comparative accuracy obtained by setting an embedding size 2 versus an embedding size 10. The lower the MAE values, the better the result.