## 3 Regularized Extreme Learning  
[PERSON] et al. [23] proposed in 2006 a learning method based on Single Hidden Layer Feedforward Neural Networks (SLFNs) that is an Extreme Learning Machine. The main idea is to randomly assign the input weight and hidden layer deviation, to set the hidden layer of the number of nodes, and then through the least squares method to determine the value of the output weight. ELM algorithm overcomes the shortcoming of local minimization and over-fitting in traditional gradient-based neural network algorithms (such as BP algorithm). Thus, the ELM is a quick speed and less intervention single hidden layer feedforward neural network. It has excellent ability of regression. In addition, ELM does not need to manually set and adjust the parameters of the classifier, the completion of the entire process without iteration. Comparing to traditional learning methods, it is easier to deploy and implement. The basic principle of the ELM is shown in Figure 3.  
Figure 2: The localization results of SM for different regional center nodes and for different number of regions (**a**) The distribution of nodes; (**b**) The distribution of anchor nodes; (**c**) The center node is #12, area is divided into three parts; (**d**) The localization result of SM corresponds to the Figure 2c; (**e**) The center node is #17, area is divided into three parts; (**f**) The localization result of SM corresponds to the Figure 2e; (**g**) The center node is #12, area is divided into two parts; (**h**) The localization result of SM corresponds to the Figure 2g. Figure 3 shows a single hidden layer feedforward neural network. Among the network there are \(m\) input nodes, \(k\) hidden layer nodes, \(m\) output nodes; \(\mathbf{\omega}_{i}\) is the input weight vector of the \(i\) hidden layer nodes, which represents the weight input of the \(j\) input node to the \(i\) hidden node; \(b_{i}\) is the deviation of the \(i\) hidden layer node (or the influence factor); \(\mathbf{\beta}\) is the weight value of the \(i\) hidden node output to the \(j\) output node; \(g(p)\) is the activation function. For the activation function, [PERSON] et al. have proved that it is an infinitely differentiable non-constant function. ELM has the characteristic of the full rank, thus it guarantees the accuracy of \(\mathbf{\omega}\) and \(\mathbf{b}\) randomly assigned. For any given input data set \((\mathbf{U},\mathbf{V})=\{(\mathbf{u}_{1},\mathbf{v}_{1}),(\mathbf{u}_{2},\mathbf{v}_{2})\,\ldots\,( \mathbf{u}_{n},\mathbf{v}_{n})\}\), the input of the \(i\) sample is \(\mathbf{u}_{i}=[u_{i1},u_{i2},\,\ldots\,\,u_{im}]^{\mathrm{T}}\in R^{m}\), and the output is \(\mathbf{v}_{i}=[\mathbf{v}_{i1},\mathbf{v}_{i2},\,\ldots\,\,\mathbf{v}_{im}]^{\mathrm{T}}\in R ^{m}\). When the \(i\) sample is input to the ELM, of the input of the \(j\) hidden layer node is shown as Equation (2):  
\[\mathbf{\omega}_{j}\mathbf{u}_{i}=\left(\omega_{j1},\omega_{j2},\cdots, \omega_{jn}\right)\left( \begin{array}{c}u_{i1}\\ u_{i2}\\ \vdots\\ u_{im}\end{array} \right) \tag{2}\] \[=\omega_{j1}u_{i1}+\omega_{j2}u_{i2}+\cdots+\omega_{jm}u_{im}.\]  
After the processing of the \(j\) hidden layer node, the output is \(g\big{(}\mathbf{\omega}_{j}\mathbf{u}_{i}+b_{j}\big{)}\). At this time, the output matrix with \(r\) hidden nodes is:  
\[\mathbf{G}(\mathbf{u}_{i})=(g(\mathbf{\omega}_{1}\mathbf{u}_{i}+b_{1}),g(\mathbf{\omega}_{2}\mathbf{u} _{i}+b_{2}),\cdots,g(\mathbf{\omega}_{r}\mathbf{u}_{i}+b_{r}))^{T} \tag{3}\]  
According to the hidden layer output matrix \(\mathbf{G}(\mathbf{X})=(\mathbf{G}(\mathbf{u}_{1}),\ldots,\mathbf{G}(\mathbf{u}_{r}))\) and the output weight matrix \(\mathbf{\beta}\), the output matrix \(\mathbf{V}\) can be approximated as:  
\[\mathbf{G}(\mathbf{u}_{i})^{T}\mathbf{\beta}=\mathbf{G}(\mathbf{u}_{i})^{T}(\mathbf{\beta}_{1},\ldots,\mathbf{\beta}_{m})\]  
\[=(g(\mathbf{\omega}_{1}\mathbf{u}_{i}+b_{1}),g(\mathbf{\omega}_{2}\mathbf{u}_{i}+b_{2}),\cdots,g(\mathbf{\omega}_{r}\mathbf{u}_{i}+b_{r}))\left(