## 1 Introduction  
Breast cancer is the most frequently diagnosed cancer among women worldwide and it is the second leading cause of death [1]. It has been shown that one woman in eight is going to develop breast cancer in her life and early diagnosis is one of the most powerful instruments we have in fighting the disease [2]. Full Field Digital Mammography (FFDM) is a non-invasive highly sensitive method for early stage breast cancer detection and diagnosis, and represents the reference imaging technique to explore the breast in a complete way [3, 4]. One of the major issues in cancer detection is due to the presence of breast dense tissue. Breast density is defined as the amount of fibrolandular parenchyma or dense tissue with respect to the fat one as seen on a mammographic exam [5]. Since X-ray absorption coefficient for dense and cancerous tissues are similar, a mammogram with a very high percentage of fibrolandular tissue is less readable. In order to have a sufficient sensitivity in denser breasts, a higher radiation dose has to be delivered to the patient [6]. Moreover, breast density is an intrinsic risk factor in developing the disease [7, 8, 9]. For these reasons, a density standard has been established by the American College of Radiology (ACR) in 2013 [5] and it is reported on the Breast Imaging Reporting and Data System (BI-RADS) Atlas. The standard defines four qualitative classes: almost entirely fatty (A), scattered areas of fibrolandular density (B), heterogeneously dense (C) and extremely dense (D). Since mammographic density assessment made by radiologists suffers from a non-negligible intra and inter-observer variability [10], some automatic methods have been developed in order to make the classification reproducible. Many approaches use a two-step classification [11; 12; 13; 14]: first, either they extract features from the images or apply a segmentation method and, afterwards, they train a classifier with a Support Vector Machine or other machine learning methods. In [15], a fully automated algorithm has been developed: the breast is segmented, density features are extracted and used to train and evaluate SVM classifiers with an accuracy of 84.47% on the miniMIAS dataset. In [16], [PERSON] et al. conceived a method based on the statistical distribution of rotationally invariant filter responses in a low dimensional space, following the Third Edition of the BI-RADS standard (1998). In the last few years, deep learning-based methods have been developed with success in a wide range of medical image analysis problems [17]. The main advantage of deep learning-based classifier stands in their capability of analyzing data from different sources and automatically extracting image related features. Since features represent image properties that cannot be analytically described, they are not easily intelligible. Moreover, it is not straightforward to explain how such algorithms perform the classification. The detailed study of deep learning applications to medical images and their explainability is a challenge that can help medical physicists on tasks such as the data quality control and validation [18]. Explaining a deep learning based classifier is crucial in order to understand whether the classification is correct. In fact, since Convolutional Neural Networks are trained directly on images, it may happen that they focus their attention on uncorrelated or wrong parts of the images [19], introducing a bias in the classification. Moreover, there exists no standard to quantify when an algorithm is well explained or not. For this reason, we propose the Spearman's rank correlation coefficient in order to verify at least the monotonic dependence between the pixel and the activation maps intensity. The lack of huge public labeled mammographic dataset is a major issue when dealing with deep machine learning models applied to mammography, because it implies the impossibility of comparing models using the same data [18].