## 1 Introduction  
One of the most important components of modern industry is rotating machinery [1]. Rolling element bearings are one of the essential parts of these machines, because they provide stiff support with a low-power consumption, thanks to their overall low friction coefficient, in a wide range of operating conditions [2]. They are also the most critical component, because they are susceptible to many failure causes not only related to wear and fatigue, but also to faulty installation, poor maintenance, and bad handling practices [3; 4; 5]. Their damage and consequent failure can cause significant unexpected downtimes, economic losses, and safety-related issues [6; 7]. Therefore, in the past few decades, industries have recognised the importance of a reliable and robust condition monitoring (CM) system for fault detection and diagnosis [8; 9]. Rolling element bearings are complex mechanical systems, in which the interaction between their elements produces vibrations even if they are \"geometrically perfect\", and the presence of faults, defects, and their location produce a characteristic impulsive vibration signature [10]. Thus, vibration analysis has become one of the most used techniques in CM, because the signal acquired from the machine contains clear fault-related signatures and can be explored easily through signal processing techniques [11; 12; 13; 14; 15; 16]. In most situations the bearing vibration cannot be measured directly, so the vibration signature is affected by noise coming from the structure and other equipment in the system. For that reason, machine users would prefer an automatic method to shorten the maintenance cycle and improve the diagnosis accuracy, which is called intelligent fault diagnosis (IFD). Thistechnique uses machine learning (ML) theories and algorithms, such as the support vector machine (SVM) [17; 18; 19; 20; 21], the k-nearest algorithm (kNN) [22; 23], artificial neural networks (ANNs) [24; 25], deep belief networks (DBNs) [26], and convolutional neural networks (CNNs) [27; 28; 29], for fault classification [30]. A database of vibration data is used to extract some important and characteristic features to train the ML algorithms at the recognition of faults. Features can be extracted from the time, frequency, or time-frequency domain, but high-dimension feature vectors may contain either redundant or irrelevant information that can increase the computational cost and reduce the accuracy of the classification algorithm [31]. Indeed, overfitting problems correlated to high-dimensional data can deteriorate the performance of ML algorithms, since their generalisability decreases as the number of features increases [32]. Thus, feature selection, with a dimensionality-reduction technique, plays a fundamental role in designing powerful and robust AI algorithms [33]. Then, unlike physics-based models, explaining the contribution given by each feature to the outcome of an IFD model is not straightforward. The term \"feature importance\" refers to a set of strategies for allocating scores to input features in a predictive model, indicating the relative significance of each item when producing a prediction [34]. Commonly used feature selection algorithms for bearing fault diagnosis include the following:  
* Filter-based methods: filters directly pre-process the collected features depending on the data intrinsic characteristics (correlation between features and class label), independent to the training of the algorithm. Typical importance indicators are Relief and Relief-F [35], information gain (IF) [36], Minimum Redundancy Maximum Relevance [37], Fischer score [38], and Distance Evaluation (DE) [39];
* Wrapper-based methods: wrappers evaluate the interaction of the feature selection activity with the training phase of classifiers [40; 41];
* Embedded methods: feature selection is performed during the classification process by adding an L1 [42] or L2 [43] regularisation term in the cost function, making the computation faster than that of the wrappers.