  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 of such divergences is the
$\gamma$-divergence and the robust estimator using the $\gamma$-divergence is
known for having a strong robustness. In this paper, we consider the robust and
sparse regression based on $\gamma$-divergence. We extend the
$\gamma$-divergence to the regression problem 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.
