## 2 Proposed Method
### Training Process of the Proposed Architecture  
The first step in the training algorithm of the proposed neural architecture is an unsupervised learning procedure that uses the FLVQ algorithm for representing each class by one or more prototypes that the procedure tries to learn from the training data in unsupervised mode, i.e., without using the known labels. The second step uses a supervised learning procedure for assessing the quality of the learned prototypes in terms of classification results. For this, the \(c\) classes are reconstructed using the nearest prototype rule, and the obtained results compared to the original classes. If one or more constructed classes differ from the original ones, the corresponding prototypes are rejected and both steps are repeated for these classes using additional competitive layers until no difference persists between the constructed and the original classes, or a maximal number of layers is reached. For this, all misclassified objects are grouped with those, if any, that present small membership degrees to all classes in order to form the training data set for an additional layer of competitive neurons. As shown in (Figure 2), the resulting architecture is a multi-competitive-layer neural network (MCLNN). Each of these layers contains \(c_{r}\) neurons representing the \(c_{r}\) detected prototypes, and \(s_{r}\) possible additional neurons representing the training examples that could not be correctly classified by this layer. These remaining examples are called \"strong elements\" and can be seen as noisy elements for the current layer. Thus, the ultimate goal of MCLNN is to represent each class by a set of one or more prototypes. The total number of the resulting prototypes for all layers of the network is \(c_{tot}=\sum_{r=1}^{q}c_{r}\), with \(c\leq c_{tot}\). During the learning process of these prototypes, whenever an object is presented to the input of a layer \(r\), all neurons of this layer are considered as winners to different degrees, and the membership degrees of the input object to the corresponding classes are computed using the expression:  
\[u_{ri}(x_{rk})=\left[\sum_{i=1}^{c_{r}}\left(\frac{\varphi_{ri}(x_{rk})}{ \varphi_{ri}(x_{rk})}\right)^{\frac{2}{n-1}}\right]^{-1} \tag{10}\]  
where \(u_{ri}(x_{rk})\) is the membership degree of the object \(x_{rk}\) to the cluster \(C_{ri}\), which is represented by the \(i\)th neuron of the \(r\)th layer; \(c_{r}\) is the number of neurons in the \(r\)th layer; \(x_{rk}\) is the \(k\)th object vector of the training data set used to train the \(r\)th layer; \(x_{rkj}\) is the \(j\)th feature of the object vector \(x_{rk}\); \(m>1\) is a weighting exponent that controls the fuzziness degree of candidate partitions during the learning process; and \(\varphi_{ri}(x_{rk})\) is the activation function of the \(i^{th}\) neuron of the \(r\)th layer, defined by the expression:  
\[\varphi_{ri}(x_{rk})=||v_{ri}-x_{rk}||^{2}=\sum_{j=1}^{p}\left(v_{rij}-x_{rkj} \right)^{2} \tag{11}\]  
where \(v_{rij}\) is the \(j\)th synaptic weight of neuron \(ri\), which is also the \(j\)th feature of prototype \(v_{ri}\) that represents cluster \(C_{ri}\); \(p\) is the number of features of the object vectors; \(i\) is the index used to locate a neuron in a competitive layer; and \(r\) is the layer's index. The first layer is trained using the original training data set \(X\), and each subsequent one with the set of remaining strong elements \(X_{r}\). Thus, at the end of the training process of each layer \(r\), all objects of its training data set \(X_{r}\) are labeled. For this, each object \(x_{rk}\) in \(X_{r}\) is assigned to the cluster \(C_{ri}\) represented by the winning prototype, which is the output neuron that maximizes \(u_{ri}(x_{rk})\) while exceeding a certain threshold \(\zeta_{r}\). The latter is introduced in this work to ensure a minimum of similarity between each processed object, and the representative prototype of the class it is assigned to, using the maximum membership degree rule.