## 2 Background
### Extreme Learning Machine  
The ELM is a learning algorithm for feed-forward artificial neural networks characterized by a single hidden layer [18; 19]. The ELM tends to minimize the training error thanks to the adoption of the smallest norm of the output weights. In addition, the ELM has a fast learning speed, as the only parameters that need to be optimized are the output weights between the hidden neurons and the output layer. The computational time required for training can be considered negligible in comparison with traditional feed-forward neural networks and support vector machines [20]. Generally, for a given training set with \(N_{m}\) instances \(\{U,T\}=\{u_{i},t_{i}\}_{i=1}^{N_{m}}\), where \(u_{i}=[u_{i1},u_{i2},...,u_{iD_{i}}]\in\mathbb{R}^{D_{i}}\) is an input sample and \(t_{i}=[t_{i1},t_{i2},...,t_{iD_{o}}]\in\mathbb{R}^{D_{o}}\) is its corresponding output (the input dimension \(D_{i}\) and output dimension \(D_{o}\) are not necessarily equal), the output of an standard single hidden layer feed forward network can be written in the following form [18; 19],  
\[\hat{t}_{k}=\sum_{i=1}^{n_{h}}\beta^{i}_{ELM}G[w_{i}\cdot u(k)+b_{i}],k=\{1,N_{ m}\}, \tag{6}\]  
where \(n_{h}\) is the number of hidden neurons, \(\beta^{i}_{ELM}\in\mathbb{R}^{1 xD_{0}}\) is the weight vector connecting the i-th neuron in the hidden layer with the output neurons, \(G\) is the activation function, \(w_{i}\in\mathbb{R}^{N_{m}x1}\) is the weight vector describing connections of the i-th hidden neuron with the input neurons, and \(b_{i}\) is the i-th hidden neuron's threshold. In the ELM algorithm, input weights \(w_{i}\) and bias \(b_{i}\) are randomly generated according to a probability distribution and are related by the inner product operator (\(\cdot\)). The training process of the ANN can be expressed as a linear regression problem for the ELM algorithm, where a zero training error between the target output and the actual output (\(\sum_{i=1}^{N_{m}}t_{k}-\hat{t}_{k}\)) can be determined as follows [18; 19]  
\[T=F\beta_{ELM}, \tag{7}\]  
where the output matrix of the hidden layer acquires the form of  
\[F= \begin{pmatrix}G(w_{1}\cdot u_{1}+b_{1})&\cdot&\cdot&\cdot&G(w_{n_{h}}\cdot u _{1}+b_{n_{h}})\\ \cdot&\cdot&\cdot\\ \cdot&G(w_{i}\cdot u_{j}+b_{i})&\cdot\\ \cdot&\cdot&\cdot&\cdot\\ G(w_{1}\cdot u_{N_{m}}+b_{1})&\cdot&\cdot&\cdot&G(w_{n_{h}}\cdot u_{N_{m}}+b_ {n_{h}})\end{pmatrix} . \tag{8}\]  
Figure 5: Single ring geometrical channel scattering model. The vehicle on the left side is the transmitter, while the vehicle on the right side represents the receiver. The optimal solution to the ELM can be computed as  
\[\beta^{*}_{ELM}=F^{\dagger}T, \tag{9}\]  
where \(F^{\dagger}\) is the Moore-Penrose generalized inverse of \(F\). If we consider zero training error, over-fitting could appear into the ELM algorithm. In order to avoid this problem, calculations are done by minimizing the final weights \(\beta_{ELM}\) with the inclusion or a regularization parameter. In other words, the mean square error is given by  
\[\min_{\beta_{ELM},e}\frac{1}{2}||\beta_{ELM}||^{2}+\frac{C}{2}\sum_{i=1}^{N_{m} }||e_{i}||^{2},\]  
\[\text{s.t}\,f(u_{i})\beta_{ELM}=t_{i}^{T}-e_{i}^{T},i=1,...,N_{m}, \tag{10}\]  
where \(f(u)\) represents an \(output(row)\) vector of matrix in expression (8), \(e_{i}\in\mathbb{R}^{D_{o}}\) is the error vector with respect to the \(i\)-\(th\) input sample, and \(C\) corresponds to the penalty coefficient, must be any real positive number. Finally, its solution when \(F\) has more rows than columns (\(n_{h}>N_{m}\)), can be written as follows [18; 19]  
\[\beta^{*}_{ELM}=\bigg{(}F^{T}F+\frac{I_{n_{h}}}{C}\bigg{)}^{-1}F^{T}T, \tag{11}\]  
where \(I_{n_{h}}\) is an identity matrix with dimension \(n_{h}\). On the other hand, in the case that \(F\) has less rows than columns (\(n_{h}<N_{M}\)), the previous solution can be written as follows  
\[\beta^{*}_{ELM}=F^{T}\bigg{(}FF^{T}+\frac{I_{N_{m}}}{C}\bigg{)}^{-1}T, \tag{12}\]  
where \(I_{N_{m}}\) is an identity matrix with dimension \(N_{m}\).