Johnson-Lindenstrauss变换在高维优化问题中的有效性及其恢复
The Effectiveness of Johnson-Lindenstrauss Transform for High Dimensional Optimization With Adversarial Outliers, and the Recovery
摘要：
本文研究高维鲁棒优化问题。由于真实世界的数据集可能包含大量的噪声，甚至是来自某些攻击者的特制样本，因此我们对具有任意（和潜在对抗性）离群值的优化问题特别感兴趣。我们专注于两个基本的优化问题：{ \em SVM与离群值}和{ \em $k$ -中心聚类与离群值}。它们实际上是极具挑战性的组合优化问题，因为我们不能对对抗性离群值施加任何限制。因此，它们的计算复杂性相当高，特别是当我们考虑高维空间中的实例时。{ \em Johnson-Lindenstrauss（JL）Transform}是最流行的降维方法之一。虽然JL变换在过去几十年中得到了广泛的研究，但其处理对抗性离群值的有效性以前从未被研究过（据我们所知）。基于一些新的见解，从几何，我们证明了这两个问题的复杂性可以显着降低通过JL变换。此外，我们证明了在降维空间的解决方案可以有效地恢复在原来的$\mathbb{R}^d$，而质量仍然保留。在实验中，我们将JL变换与其他几种常用的降维方法进行了比较，并研究了它们在合成数据集和真实数据集上的性能。
英文摘要：
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.
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