  In this paper, we consider robust optimization problems in high dimensions.
Because a real-world dataset may contain significant noise or even specially
crafted samples from some attacker, we are particularly interested in the
optimization problems with arbitrary (and potentially adversarial) outliers. We
focus on two fundamental optimization problems: {\em SVM with outliers} and
{\em $k$-center clustering with outliers}. They are in fact extremely
challenging combinatorial optimization problems, since we cannot impose any
restriction on the adversarial outliers. Therefore, their computational
complexities are quite high especially when we consider the instances in high
dimensional spaces. The {\em Johnson-Lindenstrauss (JL) Transform} 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
adversarial outliers has never been investigated before (to the best of our
knowledge). Based on some novel insights from the geometry, we prove that the
complexities of these two problems can be significantly reduced through the JL
transform. Moreover, we prove that the solution in the dimensionality-reduced
space can be efficiently recovered in the original $\mathbb{R}^d$ while the
quality is still preserved. In the experiments, we compare JL transform with
several other well known dimension reduction methods, and study their
performances on synthetic and real datasets.

Computational Geometry;Machine Learning