Gap-Free Clustering: Sensitivity and Robustness of SDP

TL;DR

This paper introduces a semidefinite programming-based clustering algorithm that robustly recovers large clusters in the stochastic block model without requiring size gaps, even with small or unrecoverable clusters present.

Contribution

The authors develop a gap-free SDP clustering method that overcomes previous limitations, providing provable recovery of large clusters regardless of small cluster sizes and noise.

Findings

Successfully recovers large clusters without size gap constraints

Achieves o(n^2) query complexity in clustering with faulty oracle

Provides novel eigenvalue perturbation bounds and leave-one-out analysis techniques

Abstract

We study graph clustering in the Stochastic Block Model (SBM) in the presence of both large clusters and small, unrecoverable clusters. Previous convex relaxation approaches achieving exact recovery do not allow any small clusters of size $o(\sqrt{n})$, or require a size gap between the smallest recovered cluster and the largest non-recovered cluster. We provide an algorithm based on semidefinite programming (SDP) which removes these requirements and provably recovers large clusters regardless of the remaining cluster sizes. Mid-sized clusters pose unique challenges to the analysis, since their proximity to the recovery threshold makes them highly sensitive to small noise perturbations and precludes a closed-form candidate solution. We develop novel techniques, including a leave-one-out-style argument which controls the correlation between SDP solutions and noise vectors even when the removal of one row of noise can drastically change the SDP solution. We also develop improved eigenvalue perturbation bounds of potential independent interest. Our results are robust to certain semirandom settings that are challenging for alternative algorithms. Using our gap-free clustering procedure, we obtain efficient algorithms for the problem of clustering with a faulty oracle with superior query complexities, notably achieving $o(n^2)$ sample complexity even in the presence of a large number of small clusters. Our gap-free clustering procedure also leads to improved algorithms for recursive clustering.