The Effectiveness of Johnson-Lindenstrauss Transform for High Dimensional Optimization With Adversarial Outliers, and the Recovery

Abstract

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.