## 2 The Framework of Deep Uniform Kernel Mapping Support Vector Machine (_D_U_KMSVM)  
SVM is a binary classification model with the learning objective to find the hyper-plane with maximum margin in the feature space so as to divide data into positive or negative classes [35]. It is well known that with the help of kernel learning, the classification performance of SVM can be further improved [10]. Given that a problem is linearly inseparable in the original space, \(\mathbf{x}^{l}\in R^{r}\) will be mapped to \(\Phi(\mathbf{x}^{l})\in H^{r^{\prime}}\) by adopting the nonlinear mapping \(\Phi:R^{r}\longrightarrow H^{r^{\prime}}(r^{\prime}>r)\), and then the theory and method in the case of linear separability are used to solve the problem. Herein, \(\Phi\) is called the kernel mapping, and is normally an unknown function. Notably, it can be implicitly induced by the kernel function \(K(\mathbf{x},\mathbf{x}^{\prime})=(\mathbf{x}\cdot\mathbf{x}^{l}+1)^{q}\), where \"\(\cdot\)\" refers to inner product. The three commonly used kernel functions are linear kernel \(K(\mathbf{x},\mathbf{x}^{i})=\mathbf{x}^{T}\cdot\mathbf{x}^{i}\), polynomial kernel \(K(\mathbf{x},\mathbf{x}^{i})=(1+\mathbf{x}^{T}\cdot\mathbf{x}^{i})^{q}\), and radial basis function \(K(\mathbf{x},\mathbf{x}^{i})=exp(-\gamma\|\mathbf{x}-\mathbf{x}^{i}|^{2})\). However, as mentioned above, the selection of kernel functions has great influence on the classification performance. A single kernel function is not applicable to all types of short text classification, while multi-kernel learning is also not competent enough to express relatively complicated compound kernel via linear or nonlinear combination of the base kernels. Therefore, to solve the kernel learning problem existing in SVM, [PERSON] et al. [34] introduced a DNMSVM. It explicitly expresses the kernel with a multi-layer perceptron, refer to [34] for more details. Motivated by DNMSVM, we present the framework of D_U_K_KMSVM, where \(U\) stands for different deep neural network, as illustrated in Figure 1.  
To explain this framework more clearly, we take multi-layer perceptron kernel mapping (DMKMSVM) as an example, as shown in Figure 2. It is equivalent to the DNMSVM model. The essence of DMKMSVM is to express the mapping kernel explicitly with a multi-layer perceptron, and then taking the output of multi-layer perceptron as the input of linear SVM, it will output the text classification results. It can be seen from Figure 2 that DMKMSVM contains two modules, namely feature extraction module and classification module. The feature extraction module defines a kernel mapping, which first represents the input data with TF-IDF, and then maps to implicit vectors using neural network, while the classification module classifies the last implicit vector using SVM. Suppose that \(W^{j}\) stands for the weight matrix between the hidden layers \(\mathbf{h}_{j-1}\) and \(\mathbf{h}_{j}\); \(\mathbf{b}^{j}\) refers to the bias vector of the layer \(\mathbf{h}_{j}\); \(\mathbf{W}^{r+1}\) and \(\mathbf{b}^{r+1}\) are the weight and bias of the output layer, respectively. For the \(l\)-th sample (\(l\) is a positive integer), the calculation process of each layer of DMKMSVM is as follows:  
\[\left\{ \begin{array}{l}\mathbf{h}_{0}^{l}=\mathbf{x}^{l},\\ \mathbf{h}_{j}^{l}=\sigma(\mathbf{W}^{l}\mathbf{h}_{j-1}^{l}+\mathbf{b}^{j}),1\leq j\leq r-1 \\ \phi(\mathbf{x}^{l})=\sigma\left(\mathbf{W}^{r}\mathbf{h}_{r-1}^{l}+b^{r}\right),\\ o^{l}=\mathbf{W}^{r+1}\phi(\mathbf{x}^{l})+b^{r+1},\end{array} \right. \tag{1}\]  
where \(\sigma\left(\cdot\right)\) is the nonlinear activation function. As a matter of fact, given the input \(\mathbf{x}^{l}\), \(\phi\left(\mathbf{x}^{l}\right)\) is a function defined by multilayer perceptron. Specifically, it will map the input to \(\phi\left(\mathbf{x}^{l}\right)\) through multiple hidden layers \(\mathbf{h}_{1}^{l}\), \(\mathbf{h}_{2}^{l}\), \(\cdots\), \(\mathbf{h}_{r-1}^{l}\).