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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 L"1 penalized least absolute deviation
method. Different from most of the other methods, the L"1 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 L"2 norm
of the estimation error is of order O(klogp/n). The result is true for a
wide range of noise distributions, even for the Cauchy distribution.
Numerical results are also presented.
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