## 3 Experiments and Results
intra-clustering quality measure equation that processes the distance of all the samples to their centroid. That is:  
\[\text{intra-clustering}=\frac{1}{|S|}\sum_{x\in S}d(x,\overline{v}) \tag{1}\]  
where:  
\[d(x,\overline{v})=\sqrt{\sum_{i\in\{1,\ldots,n\}}(x_{i}-\overline{v}_{i})^{2}} \tag{2}\]  
\(S\) is the set of samples, \(\overline{v}\) is the \(S\) centroid and '\(n\)' is the dimension size. Please note that whereas in the clustering field we look for low intra-clustering values, our embedding visualization aim is to spread embedding representations and to avoid them too being packed together. In this way, we will be able to better catch relations among samples. So, the higher our 'intra-clustering' quality measure, the better the results. In the CF embedding visualization field, we could call this quality measure an 'unpacking measure'. Table 3 shows the comparative results that test the non-variational dot product DeepMF baseline versus the proposed variational Euclidean model. Table 3 provides quality results for both user embeddings and item embeddings. As can be seen, representative improvements are obtained when the proposed model is used. Experiment 3: This experiment visually shows the improvement obtained by combining the three contributions stated in the preceding section. The visual results of the proposed variational Euclidean model have been compared to the proposed non-variational dot product baseline (DeepMF). Figure 5 shows the returned graphs for both models when applied to the datasets in Table 1. The top graphs in Figure 5 show the baseline results, whereas the bottom graphs plot the proposed model results. As can be seen, the MovieLens 100K dataset (left graphs) displays an unpacked (extended) vision of both user and item samples when the proposed model (bottom-left graph) is used, compared to the baseline (top-left graph) one. The proposed model makes it easier to compare the relationship between samples by visually inspecting the (Euclidean) distances in the graphs. It also decreases intersections between users and items embedding representations. What we are looking at here explains the 'unpacked' quality values shown in Table 3. MovieLens 1M (center graphs) and Netflix* (right graphs) show similar layouts to MovieLens 100K, suggesting that, on CF datasets, the proposed model performs as expected. As an example of the proposed model application, Figure 6 shows some demographic information from MovieLens 100K. Both graphs in Figure 6 show the location of the users. The graph on the left plots gender information: female (red) versus male (blue). The right graph plots age information: over 40 years of age (red) versus younger users (blue). Please note that the user plot in the bottom left graph of Figure 5 (MovieLens 100K) is not the same as the shapes shown in Figure 6; this is because they belong to different model trainings. Figure 6 is just an example that shows some type of demographic information: male versus female, and younger versus older users. Similar graphs can be obtained from different demographic features of users and from the item's type: zip code, incomings, educational level, genre of movies, type of music, year of book publication, etc. Figure 6 shows that there is not a clear pattern to cluster users attending to their gender or age; that is, in the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Embedding**} & \multicolumn{3}{c}{**Users**} & \multicolumn{3}{c}{**Items**} \\ \cline{2-7}  & **Proposed** & **Baseline** & **Improv.** & **Proposed** & **Baseline** & **Improv.** \\ \hline Movielens 100K & 0.8317 & 0.4789 & 73.66\% & 0.8943 & 0.6346 & 40.92\% \\ MovieLens 1M & 0.8289 & 0.5552 & 49.29\% & 0.9762 & 0.7372 & 32.42\% \\ Netflix* & 0.8790 & 0.4572 & 92.25\% & 0.9160 & 0.8569 & 6.90\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Unpacking quality measure results (intra-clustering results from the quality measure defined in the ‘Experiment 2’) for both user and item embeddings. The higher the quality value, the better the result.