## 2 Materials and Methods
### Extreme Learning Machine  
For \(N\) arbitrarily distinct samples of data \((\mathbf{x}_{i},\mathbf{t}_{i})\in\mathbf{R}^{n}\times\mathbf{R}^{m}\), where \(\mathbf{x}_{i}=[x_{i1},x_{i2},\ldots,x_{in}]^{T}\in\mathbf{R}^{n}\), \(\mathbf{t}_{i}=[t_{i1},t_{i2},\ldots,t_{im}]^{T}\in\mathbf{R}^{m}\), and an infinitely differentiable activation function of any finite interval, \(g:R\rightarrow\mathbf{R}\), standard single hidden layer feed-forward networks (SLFNs) with \(L\) hidden nodes are mathematically modeled as  
\[y(\mathbf{x}_{j})=\sum_{i=1}^{L}\beta_{i}g(\mathbf{x}_{j}\cdot\mathbf{w}_{i}+b _{i})=\mathbf{o}_{j},j=1,2,\ldots,N \tag{1}\]  
where \(\mathbf{w}_{i}=[w_{i1},w_{i2},\ldots,w_{in}]^{\mathsf{T}}\) is the connecting weight matrices of the \(i^{\text{th}}\) hidden node and the input nodes; \(\beta_{i}=[\beta_{i1},\beta_{i2},\ldots,\beta_{in}]^{\mathsf{T}}\) is the connecting weight matrices of the \(i^{\text{th}}\) hidden node and the output nodes; and \(b_{i}\) is the threshold of the \(i^{\text{th}}\) hidden node. \(\mathbf{x}_{j}\cdot\mathbf{w}_{i}\) is the inner product of \(\mathbf{x}_{j}\) and \(\mathbf{w}_{i}\). The network topology with linear output nodes is shown in Figure 1.  
Figure 1: Network topology of a standard single hidden layer feed-forward network (SLFN). Using the initialized random assignment weights \(\mathbf{w}_{i}\in\mathbf{R}^{n}\) and the thresholds \(\mathbf{b}_{i}\in\mathbf{R}\), the standard SLFNs can approximate these \(N\) samples with zero error means that \(\sum\limits_{j=1}^{N}\left\|o_{j}-t_{j}\right\|=0\), i.e., there exist \(\beta_{i}\), \(\mathbf{w}_{i}\) and \(b_{i}\), such that  
\[h(\mathbf{x}_{j})=\sum\limits_{i=1}^{L}\beta_{i}g(\mathbf{x}_{j}\cdot\mathbf{ w}_{i}+b_{i})=\mathbf{t}_{j},j=1,2,\ldots,N \tag{2}\]  
Compact form of Equation (2) can be expressed in matrices as follows:  
\[\mathbf{H}\beta=\mathbf{T} \tag{3}\]  
where,  
\[\mathbf{H}=\left[ \begin{array}{c}h(\mathbf{x}_{1})\\ \vdots\\ h(\mathbf{x}_{N})\end{array} \right]=\left[ \begin{array}{ccc}g(\mathbf{x}_{1 }\cdot\mathbf{w}_{1}+b_{1})&\ldots&g(\mathbf{x}_{1}\cdot\mathbf{w}_{L}+b_{L}) \\ \vdots&\ddots&\vdots\\ g(\mathbf{x}_{N}\cdot\mathbf{w}_{1}+b_{1})&\ldots&g(\mathbf{x}_{N}\cdot \mathbf{w}_{L}+b_{L})\end{array} \right]_{N\times L} \tag{4}\]  
\[\beta=\left[ \begin{array}{c}\beta_{1}^{\mathrm{T}}\\ \beta_{2}^{\mathrm{T}}\\ \vdots\\ \beta_{L}^{\mathrm{T}}\end{array} \right]_{L\times m} \tag{5}\]  
and  
\[\mathbf{T}=\left[ \begin{array}{c}\mathbf{t}_{1}^{\mathrm{T}}\\ \mathbf{t}_{2}^{\mathrm{T}}\\ \vdots\\ \mathbf{t}_{N}^{\mathrm{T}}\end{array} \right]_{N\times m} \tag{6}\]  
As named in Huang [19], \(\mathbf{H}\) is called the hidden layer output matrix of the neural network; The \(i^{\text{th}}\) column of \(\mathbf{H}\) is the \(i^{\text{th}}\) hidden node output with respect to inputs \(x_{1},x_{2},\ldots,x_{N}\); The row of matrix \(\mathbf{H}\) represents the hidden layer feature mapping with respect to input \(x_{i}\), that is \(x_{i}:h(x_{i})\). If the activation function \(g\) is infinitely differentiable and the nodes with parameters of hidden layers can be randomly generated, then there is [6] Theorem 1. Given an small positive value \(\varepsilon\) (\(\varepsilon>0\)), an activation function \(g\): R\(\rightarrow\)\(\mathbf{R}\) which is infinitely differentiable in any interval and, N arbitrary distinct samples \((x_{i},t_{i})\in R^{n}\times R^{m}\), there exists \(L\leq N\) for any parameters of the network \(\left\{(\mathbf{w}_{i},b_{i})\right\}_{i=1}^{L}\), according to any continuous probability distribution, then with probability one, \(\left\|\mathbf{H}_{N\times L}\beta_{L\times m}-\mathbf{T}_{N\times m}\right\|<\varepsilon\). From the point of view of interpolation, the largest number of hidden layer nodes \(L\) should be less than the number of training samples \(N\). In fact, when \(L\) is equal to \(N\), the training error will be zero.