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For the multiclass classification problem of microarray data, a new optimization model named multinomial regression with the elastic net penalty was proposed in this paper. By combining the multinomial likeliyhood loss and the multiclass elastic net penalty, the optimization model was constructed, which was proved to encourage a grouping effect in gene selection for multiclass classification.

Support vector machine [

Besides improving the accuracy, another challenge for the multiclass classification problem of microarray data is how to select the key genes [

Note that the logistic loss function not only has good statistical significance but also is second order differentiable. Hence, the regularized logistic regression optimization models have been successfully applied to binary classification problem [

For the multiclass classification of the microarray data, this paper combined the multinomial likelihood loss function having explicit probability meanings [

Given a training data set of

For the binary classification problem, the class labels are assumed to belong to

In this paper, we pay attention to the multiclass classification problems, which imply that

Following the idea of sparse multinomial regression [

In order to improve the performance of gene selection, the following elastic net penalty for the multiclass classification problem was proposed in [

For the microarray classification, it is very important to identify the related gene in groups. In the section, we will prove that the multinomial regression with elastic net penalty can encourage a grouping effect in gene selection. To this end, we must first prove the inequality shown in Theorem

Let

Note that the inequality

Hence, inequality (

Using the results in Theorem

Give the training data set

First of all, we construct the new parameter pairs

According to the inequality shown in Theorem

Microarray is the typical small

By combining the multinomial likelihood loss function having explicit probability meanings with the multiclass elastic net penalty selecting genes in groups, the multinomial regression with elastic net penalty for the multiclass classification problem of microarray data was proposed in this paper. The proposed multinomial regression is proved to encourage a grouping effect in gene selection. In the next work, we will apply this optimization model to the real microarray data and verify the specific biological significance.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by Natural Science Foundation of China (61203293, 61374079), Key Scientific and Technological Project of Henan Province (122102210131, 122102210132), Program for Science and Technology Innovation Talents in Universities of Henan Province (13HASTIT040), Foundation and Advanced Technology Research Program of Henan Province (132300410389, 132300410390, 122300410414, and 132300410432), Foundation of Henan Educational Committee (13A120524), and Henan Higher School Funding Scheme for Young Teachers (2012GGJS-063).