## 2 Methods
### \(L_{1/2}\) Regularized Logistic Regression  
Despite the advantages of \(L_{1}\) and \(L_{EN}\), there are some limitations. \(L_{1}\) and \(L_{EN}\) have a biased gene selection, and they do not have an oracle property. Besides, theoretically, the \(L_{q}\)-type regularization \(p(\beta)=\sum|\beta|^{q}\) with the lower value of \(q\) would lead to better solutions with more sparsity. However, difficulties with convergence arise when \(q\) is very close to zero. Therefore, [PERSON] et al. proposed \(L_{1/2}\) regularization. When \(\frac{1}{2}<q<1\), comparing with \(L_{1}\), the convergence of \(L_{1/2}\) regularization is not high, while when \(0<q<\frac{1}{2}\), comparing with \(L_{0}\), solving the \(L_{1/2}\) regularization is much simpler. Thus, the \(L_{1/2}\) regularization can be taken as a representative of \(L_{q}(0<q<1)\) regularization. The \(L_{1/2}\) regularized logistic regression is as follows:  
\[\beta=argmin\{l(\beta)+\lambda\sum_{j=1}^{p}|\beta_{j}|^{\frac{1}{2}}\}, \tag{7}\]  
where the value of \(\beta\) can be obtained by calculating Equation (7). In this paper, we apply the coordinate descent algorithm to solve Equation (7). The algorithm is a \"one-at-a-time\" algorithm and solves \(\beta_{j}\), and other \(\beta_{j\
eq k}\) (representing the parameters remaining after the \(j\)th element is removed) are fixed [7,8]. Suppose that we have \(n\) samples, \(D\) = (\(X_{1}\), \(y_{1}\)), (\(X_{2}\), \(y_{2}\)),..., (\(X_{n}\), \(y_{n}\)), where \(X_{i}\) = (\(x_{i1}\), \(x_{i2}\),..., \(x_{ip}\)) is the \(i\)th input pattern with dimensionality \(p\), which means the \(X_{i}\) has \(p\) genes and \(x_{ij}\) denotes the value of genes \(j\) for the \(i\)th sample. \(y_{i}\) is a corresponding variable that takes a value of 0 or 1. \(y_{i}=0\) indicates that the \(i\)th sample is in Class 1 and \(y_{i}=1\) indicates that the \(i\)thsample is in Class 2. Inspired by [PERSON] et al. [27], [PERSON] et al. [26], and [PERSON] et al. [28], the univariate half thresholding operator for a \(L_{1/2}\)-penalized logistic regression coefficient is as follows:  
\[\beta_{j}=Half(w_{j},\lambda)= \begin{cases}\frac{2}{3}w_{j}(1+\cos\frac{2(\pi- \phi_{\lambda}(w_{j}))}{3})&\text{if }|w_{j}|>\frac{3}{4}(\lambda)^{\frac{2}{3}}\\ 0&\text{if }\textit{otherwise}.\end{cases}  \tag{8}\]  
Besides, the univariate thresholding operator of the coordinate descent algorithm for the \(L_{EN}\) regularization can be defined as:  
\[\beta_{j}=f_{L_{EN}}(w_{j},\lambda,a)=\frac{S(w_{j},\lambda a)}{1+\lambda(1-a )}, \tag{9}\]  
where \(S(w_{j},\lambda a)\) is a soft thresholding operator for the \(L_{1}\) if a is equal to 1, as follows:  
\[\beta_{j}=Soft(w_{j},\lambda)= \begin{cases}w_{j}+\lambda&\text{if }w_{j}<- \lambda\\ w_{j}-\lambda&\text{if }w_{j}>\lambda\\ 0&\text{if }-\lambda\leq w_{j}\leq\lambda.\end{cases}  \tag{10}\]  
Inspired by Reference [7], Equation (7) is linearized by one-term Taylor series expansion:  
\[L(\beta,\lambda)\approx\frac{1}{2n}\sum_{i=1}^{n}(Z_{i}-X_{i}\beta)^{{}^{ \prime}}W_{i}(Z_{i}-X_{i}\beta)+\lambda\sum_{j=1}^{n}|\beta|^{\frac{1}{2}}, \tag{11}\]  
where \(Z_{i}=X_{i}\widetilde{\beta}+\frac{Y_{i}-f(X_{i}\widetilde{\beta})}{f(X_{i} \widetilde{\beta})(1-f(X_{i}\widetilde{\beta}))}\), \(W_{i}=f(X_{i}\widetilde{\beta})(1-f(X_{i}\widetilde{\beta}))\), and \(f(X_{i}\widetilde{\beta})=\frac{exp(X_{i}\widetilde{\beta})}{(1+exp(X_{i} \widetilde{\beta}))}\). Redefine the partial residual for fitting \(\widetilde{\beta}_{j}\) as \(\widetilde{Z}_{i}^{(j)}=\sum_{i=1}^{n}W_{i}(\widetilde{Z}_{i}-\sum_{k\
eq j}x _{ik}\widetilde{\beta}_{k})\) and \(\sum_{i=1}^{n}x_{ij}(Z_{i}-\widetilde{Z}_{i}^{(j)})\). A pseudocode of coordinate descent algorithm for \(L_{1/2}\) penalized logistic regression is described in Algorithm 1[7].