Robust and Sparse Regression via γ-Divergence

In high-dimensional data, many sparse regression methods have been proposed. However, they may not be robust against outliers. Recently, the use of density power weight has been studied for robust parameter estimation, and the corresponding divergences have been discussed. One such divergence is the gamma-divergence, and the robust estimator using the gamma-divergence is known for having a strong robustness. In this paper, we extend the gamma-divergence to the regression problem, consider the robust and sparse regression based on the gamma-divergence and show that it has a strong robustness under heavy contamination even when outliers are heterogeneous. The loss function is constructed by an empirical estimate of the gamma-divergence with sparse regularization, and the parameter estimate is defined as the minimizer of the loss function. To obtain the robust and sparse estimate, we propose an efficient update algorithm, which has a monotone decreasing property of the loss function. Particularly, we discuss a linear regression problem with L-1 regularization in detail. In numerical experiments and real data analyses, we see that the proposed method outperforms past robust and sparse methods.