Spectral Robustness for Correlation Clustering Reconstruction in Semi-Adversarial Models

TL;DR

This paper introduces a semi-adversarial model for correlation clustering, demonstrating that spectral algorithms are optimal for reconstruction in the pre-adversarial setting, surpassing previous limitations and revealing their robustness.

Contribution

It presents a novel pre-adversarial model and proves spectral algorithms are optimal for cluster reconstruction, extending understanding of their robustness in noisy, adversarial environments.

Findings

Spectral algorithms achieve optimal reconstruction at the information-theoretic threshold.

Spectral methods outperform SDP-based algorithms in the pre-adversarial setting.

The robustness of spectral algorithms is greater than previously believed.

Abstract

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.