# Prediction of Cell-Penetrating Peptides Using a Novel HSIC-Based Multiview TSK Fuzzy System
## 2 Materials and Methods
### Multiview TSK Fuzzy System via HSIC  
The complementarity principle is an important criterion in multiview learning. In our data, each view corresponds to a group of features, so each view has unique information. Therefore, making accurate predictions requires integrating information from each view. In this paper, we apply the Hilbert-Schmidt independence criterion (HSIC) to realize the idea of the complementarity principle. The HSIC is used to measure the independence between different views. The independence of each view can reduce redundant information and enhance complementarity. According to the method of [PERSON] et al. [19], the empirical version of HSIC is summarized as follows:  
\[\text{HSIC}\left(\mathbf{E}^{\triangleright},\mathbf{E}^{h}\right)=\left(n-1 \right)^{-2}\text{Tr}\left(\mathbf{K}^{\triangleright}\mathbf{H}\mathbf{K}^{h} \mathbf{H}\right) \tag{16}\]  
where \(\mathbf{E}^{\triangleright}\) is the prediction error in view \(v\) and \(\mathbf{K}^{v}\) is the Gram matrix in view \(v\). We set \(\mathbf{K}^{\triangleright}=\mathbf{E}^{\triangleright}(\mathbf{E}^{\triangleright })^{T}\). \(h_{ij}=\delta_{ij}-1/n\) centers the Gram matrix to have a zero mean in the feature space. For notational convenience, we ignore the scaling factor \(\left(n-1\right)^{-2}\). When all views except view \(v\) are fixed, we minimize the following function:  
\[\sum_{h=1;h\
eq v}^{V}\text{HSIC}\left(\mathbf{E}^{\triangleright},\mathbf{E}^{ h}\right)=\text{Tr}\left(\sum_{h=1;h\
eq v}^{V}\mathbf{H}\mathbf{K}^{ \triangleright}\mathbf{H}\mathbf{K}^{h}\right) \tag{17}\]  
\[=\text{Tr}\left(\sum_{h=1;h\
eq v}^{V}\mathbf{(E}^{\triangleright})^{T}\mathbf{ H}\mathbf{K}^{h}\mathbf{H}\mathbf{E}^{\triangleright}\right)=\text{Tr}\left( \mathbf{(E}^{\triangleright})^{T}\mathbf{G}^{\triangleright}\mathbf{E}^{\triangleright }\right) \tag{18}\]  
where  
\[\mathbf{G}^{\triangleright}=\sum_{h=1;h\
eq v}^{V}\mathbf{H}\mathbf{K}^{h} \mathbf{H} \tag{19}\]With the HSIC, we obtain the following objective function:  
\[\underset{\mathbf{P}^{v},\mathbf{E}^{v}}{\min}I_{\text{TSK}-\text{ HSIC}}(\mathbf{P}^{v},\mathbf{E}^{v})=\text{Tr}\Bigg{(}\frac{1}{2}\sum_{v=1}^{V} {(\mathbf{P}^{v})^{T}\mathbf{P}^{v}}+\frac{\lambda_{\mathbf{P}}^{v}}{2}\sum_{v =1}^{V}{(\mathbf{E}^{v})^{T}\mathbf{E}^{v}}+\frac{\gamma}{2}\sum_{v=1}^{V}{( \mathbf{E}^{v})^{T}\mathbf{G}^{v}\mathbf{E}^{v}}\Bigg{)} \tag{20}\]  
\[\text{s.t.}\mathbf{Y}_{vec}=\mathbf{X}_{S}^{v}\mathbf{P}^{v}+\mathbf{E}^{v}\text { for }v=1,2,\ldots,V \tag{21}\]  
where \(\mathbf{X}_{S}^{v}\) is the matrix of the input data of view \(v\) through the transformation of Equation (10), \(\mathbf{Y}_{vec}\) is the matrix obtained from the vector through one-hot coding, \(\mathbf{E}^{v}\) represents the error matrix of view \(v\), \(\mathbf{Y}_{vec},\ \mathbf{E}^{v}\in\mathbf{R}^{N\times C}\), and \(\mathbf{P}^{v}\) is a matrix composed of the consequent parameters of the TSK fuzzy system, \(\mathbf{P}^{v}\in\mathbf{R}^{K(D^{v}+1)\times C}\). \(V\) is the number of all views, \(C\) is the total number of classes, \(N\) is the total number of data samples, \(K\) is the total number of rules, and \(D\) is the number of data dimensions. \(\gamma,\lambda_{\mathbf{P}}^{v}\) are regularization parameters. Their values can be obtained by cross validation. The Lagrange function of this problem is defined as:  
\[\mathcal{L}(\mathbf{P}^{v},\mathbf{E}^{v})=I_{\text{TSK}-\text{ HSIC}}(\mathbf{P}^{v},\mathbf{E}^{v})-\text{Tr}(\sum_{v=1}^{V}{\mathbf{\alpha}^{v} \Gamma}{\left(\mathbf{X}_{S}^{v}\mathbf{P}^{v}+\mathbf{E}^{v}-\mathbf{Y}_{vec }\right)}) \tag{22}\]  
\[\text{Let }\partial\mathcal{L}/\partial\mathbf{P}^{v}=0,\partial\mathcal{L}/ \partial\mathbf{E}^{v}=0\text{ and }\partial\mathcal{L}/\partial\mathbf{\alpha}^{v}=0\text{:}\]  
\[\left\{