## 1 Introduction  
With the development of DNA microarray technology, biological researchers can pay more attention to simultaneously studying the expression levels of thousands of genes [1; 2]. Cancer classification based on gene expression levels is one of the most active topics in genome research, which is appropriate for gene expression levels in different situations (e.g., normal and abnormal) [3; 4]. However, cancer classification using DNA microarray data is a challenge because of the data's high dimension and small sample size [5]. Generally, the number of genes ranges in the thousands from a hundred or fewer tissue samples, and so gene selection has recently emerged as important technology for cancer classification [6]. Gene selection is applied because only a small subset of genes is strongly indicative of a targeted disease. From the biological perspective, effective gene selection methods can be desirable to help to classify different types of cancer and improve the accuracy of prediction [7; 8; 9]. Many gene selection methods have been proposed for selection of the subset of meaningful and important genes that can achieve high cancer classification performance. Recently, there has been growing interest in applying regularization techniques in gene selection. Regularization methods are an important embedded technique [10; 11; 12; 13]. From the statistical perspective, regularization methods can prevent over-fitting. Many statistical methods have been successfully applied to cancer classification. Among them, logistic regression [14; 15; 16; 17] is a powerful discriminative method, and has a direct probabilistic interpretation that can obtain classification probabilities apart from the class label information. However, logistic regression is not suitable for solving the high-dimensional and small sample size problem because the design matrix is singular. Thus, [PERSON]'s method cannot work. Regularized logistic regression has been successfully applied in cancer classification in order to be suitable for high dimension and small sample size [7; 8]. The advantages of regularized logistic regression can improve the classification accuracy by shrinking the regression coefficients and selecting a small subset of genes. Different regularization terms are applied to regularized logistic regression. The widely popular regularization term is \(L_{1}\) penalty, which is the least absolute shrinkage and selection operator (lasso) [18]. Meanwhile, there are various of versions of \(L_{1}\), such as smoothly clipped absolute deviation (SCAD) [19], maximum concave penalty (MCP) [20], group lasso [21], and so on. The \(L_{1}\) regularization can assign some genes' coefficients to zero for variable selection. Thus, the \(L_{1}\) regularization has been widely applied to data with high dimension and small sample size. Although a well-known regularization method is the \(L_{1}\) penalty, it has some limitations [22]. The \(L_{1}\) regularization does not have oracle property [19], which means the aim-listed probability of selecting the right set of genes (with nonzero coefficients) converges to one, and the estimators of the nonzero coefficients have asymptotically normal distribution with the same means and covariances as if the zero coefficients were known in the prior. Besides, there is grouping among genes in DNA microarray data. Related to this limitation, concerning the grouping property, [PERSON] and [PERSON] proposed the elastic net penalty (\(L_{EN}\)) [23], which is a linear combination of \(L_{1}\) and \(L_{2}\) penalties. In addition, \(L_{1}\) regularization is not sparser. To overcome this limitation, [PERSON] et al. proposed the \(L_{1/2}\) penalty--a method that can be taken as a representative of \(L_{q}\) (\(0<q<1\)) penalty in both sparsity and computational efficiency, and has demonstrated many attractive properties, such as unbiasedness and oracle properties [24; 25; 26]. Therefore, we investigated \(L_{1/2}\) regularized logistic regression for gene selection in cancer classification.