Simultaneous variable selection and class fusion with penalized distance criterion based classifiers

Two new methods are proposed to solve the problem of constructing multiclass classifiers, selecting important variables for classification and determining corresponding discriminative variables for each pair of classes simultaneously in the high-dimensional setting. Different from existing methods, which are based on the separate estimation of the precision matrix and mean vectors, the proposed methods construct classifiers by estimating products of the precision matrix and mean vectors or all discriminant directions directly with appropriate penalties. This leads to the use of the distance criterion instead of the log-likelihood used in the existing literature. The proposed methods can not only consistently select important variables for classification but also consistently determine corresponding discriminative variables for each pair of classes. For the multiclass classification problem, conditional misclassification error rates of classifiers constructed by the proposed methods converge to the misclassification error rate of the Bayes rule in probability and rates of convergence are also obtained. Finally, simulations and the real data analysis well demonstrate good performances of the proposed methods in comparison with existing methods.