## 5 Discussion
### Explainable AI in Breast Cancer Classification  
Concerning the mathematical explanation, as emerged from the visualization of the feature embeddings, one can discern that both t-SNE and UMAP are able to extract meaningful relationship in the low-dimensionality spaces when the features are representative of the underlying patterns in the sample images. In Figures 9 and 10, four clusters are clearly visible for the DenseNet-161 architecture. On the contrary, when the model is less accurate, as in the case of fast and light SqueezeNet (in terms of number of parameters), the cluster formation behaves differently, with UMAP resulting in more compact representations. As a general suggestion, therefore, the study recommends to use these mathematical XAI techniques to visualize if considered features for a problem under consideration are relevant. With respect to the perceptive XAI techniques, the performance results of the CNN models are aligned with the complementary information that can be extracted from Grad-CAM and LIME methods. While the first allows to detect which regions have a gradient that is deemed relevant for performing the prediction, the second permits to understand, for each superpixel, if it is positively or negatively correlated to the prediction. Moreover, the LIME method has an adjustable parameter for deciding the number of top contributing features to show over the original image. Since we already have the intensity values from the saliency maps of Grad-CAM, we decided to mark in green every positively correlated region and in red every negatively correlated region, so that the mixed information obtained can be exploited to obtain an intuitive understanding of which regions are more important (higher intensity values in CAM maps), and which are positively or negatively correlated to the final outcome (green and red, respectively). Interestingly, the CNN architectures that find trouble in correctly identifying the lesion areas also appear to have a lower AUC. Thus, on a general scale, the higher AUC can be explained by using XAI methods. For instance, the SqueezeNet, which is the worst-performing network in terms of AUC and validation loss, and VGG-16, which has a trade-off between the AUC and the number of parameters, as shown in Figure 8, fail to spotlight the relevant lesions as illustrated in Figure 11. In contrast, DenseNet-161, DenseNet-121, ResNeXt, and ResNet-50, which feature higher AUC values, correctly highlight the lesion when tested with the Grad-CAM method. Moreover, as the loss trends and the AUC tables show, none of the CNNs yielded 100% performance, which means the misclassified examples are also present. These samples of the misclassified images are also presented to the XAI methods in order to dive into the features that resulted in misclassification. The reason behind misclassification of one type of cancerous image to another type might be related to the homogeneity of the shapes of a few examples with other classes. Figure 13 illustrates the results of both Grad-CAM and LIME methods regarding examples of misclassified images. The labels provided above the samples represent the ground truth, whereas the labels provided under the saliency maps are the predictions made by the CNNs. This figure proves that XAI could also help the physician understand why the AI is failing. For instance, the _None_ image in the Figure 13 contains a mesh that is not lesion according to the expert radiologists. However, it fools the CNN to misclassify the image as _Ori_. Both the CAM and LIME methods highlighted the regions that carry analogous properties, thus explaining the cause of the misclassification. It is worth noting that a similar discussion emerges from the other examples provided in Figure 13.  
In order to understand how the results of two XAI perceptive methods vary according to the different target classes, Figure 14 reports the explanation results of both methods considering eight different correctly classified images.