## 2 Background
### Semi-Supervised Extreme Learning Machine  
In a semi supervised setting, we have few labeled data and plenty of unlabeled data, which can be used to increase the perform of the system [18]. For this approach, we assume that the distribution of the label data is similar to the distribution of the unlabeled data and try to extract that the beneficial information. The SS-ELM algorithm is presented in Algorithm 2. ```
Inputs: Labeled data \(\{U_{I},T_{I}\}=\{u_{i},t_{i}\}_{i=1}^{l}\), where \(u_{i}=[u_{i1},u_{2},...,u_{iD_{i}}]\in\mathbb{R}^{D_{i}}\) and \(t_{i}=[t_{i1},t_{i2},...,t_{iD_{i}}]\in\mathbb{R}^{D_{o}}\), and unlabeled data \(U_{I\mu}=\{u_{i}\}_{i=1}^{W}\), where \(l\) and \(u\) are the numbers of labeled and unlabeled data, respectively Output: The output weights \(\beta_{ELM}\)
1: Construct the graph Laplacian \(L\) from both \(U_{I}\) and \(U_{\mu}\)
2: Initiate an ELM network of \(n_{h}\) hidden neurons with random input weights and biases, calculate the output matrix of the hidden neurons \(F\)
3: Select the hyper-parameters \(C\) and \(\lambda\)
4: Find the output weights if\(n_{h}<=N_{m}\)then

\(\beta_{ELM}=(I_{n_{h}}+F^{T}CF+\lambda F^{T}L)^{-1}F^{T}C\widetilde{T}\), where \(\widetilde{T}\in I\mathbb{R}^{(l+\mu)xD_{o}}\) is the augmented training target with its first \(l\) rows equal to \(T_{I}\) and the rest equal to \(0\), \(C\in I\mathbb{R}^{(l+\mu)x(l+\mu)}\) is a diagonal matrix with \([C]_{ii}=C\), \(i=1,...,l\) and the rest of values equal to \(0\) else \(\beta_{ELM}=F^{T}(I_{I+\mu}+CFF^{T}+\lambda LFF^{T})^{-1}C\widetilde{T}\), where \(I_{I+\mu}\) is an identity matrix of dimension \(l+\mu\) end
```  
**Algorithm 2**SS-ELM algorithm. It is important to notice that \(\lambda\) is known as the trade-off parameter, and indicates how important the unlabeled data are in the training stage. Obviously, if we set this parameter to zero, then the SS-ELM matches the standard ELM. The Laplacian graph is used to extract the geometry distribution information contained in the available data. As previously indicated, one of the assumptions of SS learning is that both labeled data \(U_{I}\) and unlabeled data \(U_{I\mu}\) are drawn from the same marginal distribution \(P_{U}\). Another assumption is that the conditional probabilities of \(P(t|u_{1})\) and \(P(t|u_{2})\) should be similar if the two points \(u_{1}\) and \(u_{2}\) are close. With these assumptions, the manifold regularization term is usually minimized with the following function [20]  
\[L_{m}=\frac{1}{2}\sum_{i,j}r_{i,j}||P(t|u_{i})-P(t|u_{j})||^{2}, \tag{13}\]  
where \(r_{i,j}\) is the pair-wise similarity between two inputs and can be computed with the next expression  
\[r_{i,j}=\left\{ \begin{array}{rl}&\exp(-||u_{i}-u_{j}||^{2}/2\sigma^{2})\text { if }u_{i},u_{j}\text{ are neighbors }\\ &0\text{ otherwise }\end{array} \right., \tag{14}\]  
Expression (13) can be simplified as \(L_{m}=Tr(\hat{T}^{T}L\hat{T})\), where \(Tr(\cdot)\) is the trace of a matrix. The Laplacian graph can be calculated as \(L=D-R\), where \(D\) is a diagonal matrix with its diagonal elements \(D_{ii}=\sum_{j=1}^{l+\mu}r_{i,j}\), and the similarity matrix \(R=[r_{i,j}]\). As mentioned, the SS-ELM setting incorporates the manifold regularization to incorporate the use of unlabeled data, thus improving the accuracy of the predictions. The expressions for \(\beta_{ELM}\) shown on Algorithm 2 are obtained when modifying the problem on Equation (10) as follows,  
\[\min_{\beta}\frac{1}{2}||\beta_{ELM}||^{2}+\frac{1}{2}\sum_{i=1}^{N_{m}}C_{i} ||e_{i}||^{2}+\frac{\lambda}{2}Tr(\beta_{ELM}^{T}F^{T}LF\beta_{ELM}),\]  
\[\text{s.t}\,f(u_{i})\beta_{ELM}=t_{i}^{T}-e_{i}^{T},i=1,...,N_{m}. \tag{15}\]  
In the context of V2V communications, this algorithm might offer a better performance, as it could, for example, learn from the time and frequency variations present in the propagation scenario.