半对抗模型中相关聚类重构的谱鲁棒性
Spectral Robustness for Correlation Clustering Reconstruction in Semi-Adversarial Models
摘要：
相关聚类是一个重要的聚类问题，有着广泛的应用。我们研究了这个问题的重建版本，其中一个是寻求重建一个潜在的聚类，已被随机噪声和对抗性修改损坏。关于后者，在文献中有一个标准的“后对抗”模型，其中对抗性修改在噪声之后。在这里，我们介绍并分析了一个“预对抗”模型，其中对抗性修改先于噪声。给定来自这种半对抗生成模型的输入，目标是几乎完美地重建潜在聚类，并且具有很高的概率。我们专注于隐藏聚类具有几乎相等大小的情况，并显示以下内容。在对抗前的环境中，谱算法是最佳的，因为它们一直重建到信息论阈值，超过该阈值就不可能重建。这与后对抗设置相反，在后对抗设置中，它们恢复隐藏聚类的能力在阈值之前停止，但基于SDP的算法最佳地填补了这一空白。这些结果突出了迄今为止未知的光谱算法的鲁棒性，表明它们比以前认为的更脆弱。
英文摘要：
Correlation Clustering is an important clustering problem with many applications. We study the reconstruction version of this problem in which one is seeking to reconstruct a latent clustering that has been corrupted by random noise and adversarial modifications. Concerning the latter, there is a standard "post-adversarial" model in the literature, in which adversarial modifications come after the noise. Here, we introduce and analyse a "pre-adversarial" model in which adversarial modifications come before the noise. Given an input coming from such a semi-adversarial generative model, the goal is to reconstruct almost perfectly and with high probability the latent clustering. We focus on the case where the hidden clusters have nearly equal size and show the following. In the pre-adversarial setting, spectral algorithms are optimal, in the sense that they reconstruct all the way to the information-theoretic threshold beyond which no reconstruction is possible. This is in contrast to the post-adversarial setting, in which their ability to restore the hidden clusters stops before the threshold, but the gap is optimally filled by SDP-based algorithms. These results highlight a heretofore unknown robustness of spectral algorithms, showing them less brittle than previously thought.
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