JDS Journal of Data Science 1683-8602 1680-743X 1680-743X School of Statistics, Renmin University of China JDS1000 10.6339/20-JDS1000 Statistical Data Science Sparse Learning with Non-convex Penalty in Multi-classification LiNan1 ZhangHao Helenhzhang@math.arizona.edu2∗ Department of Epidemiology and Cancer Control, St. Jude Children’s Research Hospital, Memphis, Tennessee, U.S.A. Department of Mathematics, University of Arizona, Tucson, Arizona, U.S.A. Corresponding author. Email: hzhang@math.arizona.edu. 20211022021 1915674 Supplementary Material

A zip file includes all the computation code and data for the numerical experiments is available.

 112020 122020 2021 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.2021 Open access article under the CC BY license.

Multi-classification is commonly encountered in data science practice, and it has broad applications in many areas such as biology, medicine, and engineering. Variable selection in multiclass problems is much more challenging than in binary classification or regression problems. In addition to estimating multiple discriminant functions for separating different classes, we need to decide which variables are important for each individual discriminant function as well as for the whole set of functions. In this paper, we address the multi-classification variable selection problem by proposing a new form of penalty, supSCAD, which first groups all the coefficients of the same variable associated with all the discriminant functions altogether and then imposes the SCAD penalty on the supnorm of each group. We apply the new penalty to both soft and hard classification and develop two new procedures: the supSCAD multinomial logistic regression and the supSCAD multi-category support vector machine. Our theoretical results show that, with a proper choice of the tuning parameter, the supSCAD multinomial logistic regression can identify the underlying sparse model consistently and enjoys oracle properties even when the dimension of predictors goes to infinity. Based on the local linear and quadratic approximation to the non-concave SCAD and nonlinear multinomial log-likelihood function, we show that the new procedures can be implemented efficiently by solving a series of linear or quadratic programming problems. Performance of the new methods is illustrated by simulation studies and real data analysis of the Small Round Blue Cell Tumors and the Semeion Handwritten Digit data sets.

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