Multinomial logistic regression classifier via [formula omitted]-proximal Newton algorithm.

Multinomial logistic regression (MLR) is a useful tool for solving multi-classification problems. The l q , 0 (q ⩾ 1) norm is an ideal regularization term for characterizing group sparsity in multinomial logistic regression and selecting important features in the high dimensional data. However, l q , 0 regularized multinomial logistic regression ( l q , 0 -MLR) is nonconvex, discontinuous, and NP-hard. Thus, most prior studies adopted a continuous approximation of the l q , 0 norm. In this paper, we present a novel l q , 0 -proximal Newton algorithm ( l q , 0 -PNA) to solve the l q , 0 -MLR. We first define a strong α -stationary point and prove that this point is a local minimizer of l q , 0 -MLR. We then convert such a point into a stationary equation and solve it by l q , 0 -PNA, which is a Newton-type method running on a group sparse subspace with a low computational cost. Furthermore, we establish a locally quadratic convergence of l q , 0 -PNA. Finally, numerical experiments on simulated and real data show the superiority of l q , 0 -PNA in terms of computational time and accuracy, when compared with six state-of-the-art solvers, especially for high dimensional data. [ABSTRACT FROM AUTHOR]