## 2 Conventional SOM, Its Selected Generalizations, and Related Work
### Conventional SOM  
Consider a conventional SOM with most often used two-dimensional, rectangular map of neurons (with \(m_{1}\times m_{2}\) neurons) as shown in Figure 1 (the black lines represent physical connections in the network whereas the green lines--non-physical, topological ones). Assume that the map has \(n\) inputs (features, attributes) \(x_{1},x_{2},\ldots,x_{n}\) and consists of \(m\) neurons (\(m=m_{1}\cdot m_{2}\)); their outputs are \(y_{1},y_{2},\ldots,y_{m}\), where \(y_{j}=\sum_{i=1}^{n}w_{ji}x_{i}\), \(j=1,2,\ldots,m\) and \(w_{ji}\) are weights connecting the \(i\)-th input of the network with the output of the \(j\)-th neuron. Using vector notation (\(\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{T}\), \(\mathbf{w}_{j}=[w_{j},w_{2},\ldots,w_{jn}]^{T}\)), \(y_{j}=\mathbf{w}_{j}^{T}\mathbf{x}\). The learning data consists of \(L\) input vectors \(\mathbf{x}_{l}\) (\(l=1,2,\ldots,L\)). Assuming that the learning vectors are normalized, the neuron \(j_{\mathbf{x}}\) which wins in the competition of neurons when the learning vector \(\mathbf{x}_{l}\) is presented to the network is selected in the following way:\[d(\mathbf{x}_{l},\mathbf{w}_{j_{x}})=\min_{j=1,2,...,m}d(\mathbf{x}_{l},\mathbf{w}_{j}), \tag{1}\]  
where \(d(\mathbf{x}_{l},\mathbf{w}_{j})\) is a distance measure between \(\mathbf{x}_{l}\) and \(\mathbf{w}_{j}\) (the most often used Euclidean measure is applied). A Winner-Takes-Most (WTM) algorithm is used in the learning process of the considered networks. The WTM learning rule is formulated as follows:  
\[\mathbf{w}_{j}(k+1)=\mathbf{w}_{j}(k)+\eta_{j}(k)N(j,j_{\mathbf{x}},k)[\mathbf{x}(k)-\mathbf{w}_{j }(k)], \tag{2}\]  
where \(k\) is the iteration number, \(\eta_{j}(k)\) is the learning coefficient, and \(N(j,j_{\mathbf{x}},k)\) is the neighborhood function of the \(j_{\mathbf{x}}\)-th winning neuron. Most often the Gaussian-type neighborhood functions are used, i.e.,:  
\[N(j,j_{\mathbf{x}},k)=e^{-\frac{d_{tpl}^{2}(j_{\mathbf{x}})}{2\lambda^{2}(k)}}, \tag{3}\]  
where \(\lambda(k)\) is the neighborhood radius and \(d_{tpl}(j,j_{\mathbf{x}})\)--the topological distance between the \(j_{\mathbf{x}}\)-th and \(j\)-th neurons. Figure 2 (Initialization-, WTM-algorithm-, and SOM-blocks) presents the block scheme of the learning algorithm of the conventional SOM. \(e\) and \(e_{max}\) in Figure 2 (and also in Figure 3) stand for epoch number and maximal number of epochs, respectively; the remaining symbols in both figures were introduced earlier in this section. In order to visualize the distance- and density structures of the learning data on two-dimensional map of neurons, the so-called U-matrix has been proposed [5]. It consists of \(m=m_{1}\cdot m_{2}\) components \(U_{j}\) (\(j=1,2,...,m\)) calculated as follows:  
\[U_{j}=\sum_{h\in S_{j}}d(\mathbf{w}_{h},\mathbf{w}_{j}), \tag{4}\]  
where \(S_{j}\) represents the neurons that are immediate neighbors of the considered \(j\)-th neuron. Therefore, a given \(U_{j}\) represents the local distance structure in the input data. The U-matrix is usually visualized by means of shades in a gray scale (the smaller value of \(U_{j}\)--the lighter shade is used and vice versa, the larger \(U_{j}\) corresponds to the darker shade). Figure 1: Conventional self-organizing map (SOM) with two-dimensional, rectangular map of neurons. Figure 2: Block schemes of the learning algorithms of the conventional SOM, GSOM, GGN, IGG, GeSOM with 1 DN (DSOM), and GeSOM with T-LSs. Figure 3: Block scheme of the learning algorithm of GNG. First, we illustrate the operation of the conventional SOM in the case of the well-known and relatively simple _Zoo_ data set available from the UCI repository of machine learning databases ([[https://archive.ics.uci.edu/ml](https://archive.ics.uci.edu/ml)]([https://archive.ics.uci.edu/ml](https://archive.ics.uci.edu/ml))). The _Zoo_ data set contains 101 records; each record is described by 16 attributes (15 Boolean and 1 numerical).