## 1 Introduction
### Broad Learning System  
The BLS is a forward neural network based on the connection of random vector functions. Its network structure is divided into a mapping layer, reinforcement layer, and output layer, as shown in Figure 1. Compared with the random vector function connectivity network, the mapping layer replaces the output layer. The BLS can update the network structure quickly by adding nodes in the mapping and enhancement layers, which benefits from the efficient incremental algorithm. The BLS extracts the mapping features of the original data by feeding the input data into the mapping layer and then uses the output of the mapping layer as the input of the enhancement layer. It can achieve the data feature and finally obtains the output weight matrix by the ridge regression method. The structure of the BLS can be flexibly changed, and the CMBLS is a typical variant. The network of CMBLS is shown in Figure 2. CMBLS cascades each node of the mapping layer in turn. It maps the original input data as the first mapping node through the mapping function. Then the output of the first mapping node is used as the second mapping node input. It continues to form the structure of the mapping layer cascade, which achieves sparse and nonlinear mapping of data features [32]. It can weaken the correlation between adjacent features and improve the transformation of low-level to high-level features. As a basis of the proposed method in this paper, the CMBLS is introduced first. As shown in Figure 2, the mapping layer node is defined as \(\mathbf{Z}=[\mathbf{Z}_{1},\mathbf{Z}_{2},\cdots,\mathbf{Z}_{n}]\), the enhancement layer node is defined as \(\mathbf{H}=[\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{m}]\), and the input data sequence is \(\mathbf{X}\). The first node of the mapping layer is \(\mathbf{Z}_{1}\). \[\mathbf{Z}_{1}=\varphi(\mathbf{X}\mathbf{W}_{\text{e1}}+\mathbf{\beta}_{\text{e1}})\triangleq \varphi(\mathbf{X};\{\mathbf{W}_{\text{e1}},\mathbf{\beta}_{\text{e1}}\}) \tag{1}\]  
where \(\mathbf{W}_{\text{e1}},\mathbf{\beta}_{\text{e1}}\) are the randomly initialized weight matrix and bias vector, respectively, \(\varphi\) is the mapping function, \(\mathbf{W}_{\text{e2}},\mathbf{\beta}_{\text{e2}}\) are similar to \(\mathbf{W}_{\text{e1}},\mathbf{\beta}_{\text{e1}}\) in that they are generated through a random initialization mechanism. The second mapping node \(\mathbf{Z}_{2}\) is defined as:  
\[ \begin{array}{ll}\mathbf{Z}_{2}&=\varphi(\mathbf{Z}_{1}\mathbf{W}_{\text{e2}}+\mathbf{\beta }_{\text{e2}})\\ &\triangleq\varphi(\varphi(\mathbf{X}\mathbf{W}_{\text{e1}}+\mathbf{\beta}_{\text{e1}})\bm {W}_{\text{e2}}+\mathbf{\beta}_{\text{e2}})\\ &\triangleq\varphi^{2}(X;\{\mathbf{W}_{\text{e1}},\mathbf{\beta}_{\text{e1}}\},\{\mathbf{ W}_{\text{e2}},\mathbf{\beta}_{\text{e2}}\})\\ &\triangleq\varphi^{2}\Big{(}X;\{\mathbf{W}_{\text{e1}},\mathbf{\beta}_{\text{e1}}\}_{ 1=2}\Big{)}\end{array}  \tag{2}\]  
Figure 1: The network structure of BLS [19]. By the same analogy, the \(k\)_th_ mapping layer node is defined as:  
\[ \begin{array}{ll}\mathbf{Z}_{k}&=\varphi(\mathbf{Z}_{k-1}\mathbf{W}_{ek}+\mathbf{\beta}_{ek})\\ &=\varphi^{n}\Big{(}\mathbf{X};\{\mathbf{W}_{ei},\mathbf{\beta}_{el}\}^{k}_{i-1}\Big{)},\, \text{for}\,\,k=1,\ldots,n\end{array}  \tag{3}\]  
where \(\mathbf{W}_{ei},\mathbf{\beta}_{el}\) is the randomly initialized weight matrix and bias vector, and the input of the enhancement layer node is the input of the mapping layer node, so the enhancement layer node \(\mathbf{H}_{i}\) is defined as:  
\[\mathbf{H}_{i}=\xi(\mathbf{Z}\mathbf{W}_{hl}+\mathbf{\beta}_{hl}),\,i=1,\ldots,m \tag{4}\]  
where \(\mathbf{W}_{hl},\mathbf{\beta}_{hl}\) is randomly initialized, \(\xi\) is the excitation function, the enhancement layer node is defined as \(\mathbf{H}=[\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{m}]\), and the combined matrix of the enhancement and mapping layers is \(\mathbf{A}=[\mathbf{Z}|\mathbf{H}]\), so the output \(\mathbf{Y}\) and the output weight matrix \(\mathbf{W}^{out}\) of CMBLS are defined as:  
\[