Spectral Robustness for Correlation Clustering Reconstruction in Semi-Adversarial Models
This work addresses the robustness of clustering algorithms in semi-adversarial scenarios, providing insights for applications in data analysis where adversarial noise is a concern, though it is incremental as it builds on existing models.
The paper tackles the problem of reconstructing latent clusterings corrupted by both random noise and adversarial modifications, introducing a 'pre-adversarial' model where adversarial changes occur before noise. It shows that spectral algorithms achieve optimal reconstruction up to the information-theoretic threshold in this model, unlike in the 'post-adversarial' setting where they fall short.
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.