  In this paper, the high-dimensional sparse linear regression model is
considered, where the overall number of variables is larger than the number of
observations. We investigate the L1 penalized least absolute deviation method.
Different from most of other methods, the L1 penalized LAD method does not need
any knowledge of standard deviation of the noises or any moment assumptions of
the noises. Our analysis shows that the method achieves near oracle
performance, i.e. with large probability, the L2 norm of the estimation error
is of order $O(\sqrt{k \log p/n})$. The result is true for a wide range of
noise distributions, even for the Cauchy distribution. Numerical results are
also presented.
