Before revision: {\em   Based on some novel insights in geometry, we prove that the complexities of these two problems can be significantly reduced through JL transform .
After revision: {\em is one of the most popular methods for dimension reduction. Though the JL transform has been widely studied in the past decades, its effectiveness for dealing with outliers has never been investigated before (to the best of our knowledge). Based on some novel insights in geometry, we prove that the complexities of these two problems can be significantly reduced through JL transform .