## 3 Results  
### Comparing the Embedding Techniques  
Our targeted roadmap towards a XAI being able to identify optimal therapy schemes has to rely on the detection of most informative variables in this purpose, whose role can be verified by the procedure adopted in current guidelines. First of all, we evaluated the different embedding techniques and compared their output using \(n=2\) dimensions, as shown in Figure 2.  
The color legend shows the molecular subtype of the patient's tumor. Interestingly, the two-dimensional embedding shows a cluster structure. Clusters seem to reproduce the molecular subtype, a property slightly affected by the embedding technique, but ADR emphasizes the separation of each group.  
To evaluate the informative content of such clusters and to obtain an interpretation, we considered which features most influence this representation. Focusing on the ADR technique, which resulted in the clearest representation, we evaluated the most important eigenvectors and examined their components: ER, PgR, Kii67 and HER2 resulted as the most significant ones. We underline that these features are exactly those defining the molecular subtypes, according to the associated decisional trees, whose selection in the relevant space appears to be an important result from a clinical point of view. By construction, the sum of squared coefficients of vectors spanning the relevant space equals to one, so we express the aforementioned weights in terms of percentages assumed by the square coefficient associated with each initial feature. The first principal component in output of the ADR procedure assigns to ER a squared coefficient equal to 20.9%, 16.1% to PgR, 2.8% to Kii67 and 56.7% to HER2. Globally, these initial features cover 96.5% of the overall unitary sum of squared coefficients. The second ADR component allocates to ER 37.5%, to PgR 18.8%, to Kii67 1.8% and to HER2 39.2%, now totally covering 97.3%, that is quite the overall variability.  
We chose some well-established dimensional reduction procedures to underline the robustness of the selected information once the same characterization is approximately observed in the output of each technique. The PCA embedding yields molecular subtype groups with a low separation with respect to remaining methods, because of the very simple structure of the associated algorithm. Nevertheless, it is possible already to underline the common indistinguishability shared by all embedding techniques between Luminal A and Luminal B HER2 negative patients. ADR, Laplacian eigenmap and DVE show much more separated HER2 positive and Luminal B HER2 positive patients, as well as triple negative cases. The absence for free parameters other than the embedded space dimension gives to ADR a privileged role for our purposes. Indeed, both Laplacian eigenmap and DVE are based on local patches defined through a parameter associated with the number of neighbors for each patient. In Figure 2 both cases adopt this number equal to 15 to maximize the cluster separation, even if it is data set dependent, because a different number of members is associated with each molecular subtype and patches should match as much as possible with this membership limitation.