Gap-Free Clustering: Sensitivity and Robustness of SDP

TL;DR

The paper develops a gap-free SDP framework for unbalanced SBM clustering, showing that large clusters can be exactly recovered without requiring a size gap between large and small clusters. It introduces a novel leave-one-out perturbation analysis and an improved eigenvalue perturbation bound to handle mid-size clusters near the recovery threshold, enabling robust, block-diagonal SDP solutions whose blocks are rank-zero or rank-one. The framework extends to semirandom perturbations, recursive clustering, and clustering with a faulty oracle, achieving improved sample complexities and instance-adaptive guarantees. Collectively, these contributions advance gap-free, robust clustering with practical implications for large-scale networks and adaptive data scenarios.

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.

Figures (1)

• Figure 1: SDP solutions and effects of regularization $\lambda$. (a): Ground truth clusters of sizes 300, 150, and 50 ($\times 5$). (b)&(c): Solutions to SDP \ref{['eq:SDP_regularized_original']} with different $\lambda$. Nonzero off-block-diagonal entries are highlighted in pink; other entries shown in grayscale ($\text{white}=0,\text{black}=1$).

Theorems & Definitions (86)

• Theorem 1: Informal version of Theorem \ref{['thm:clustering_with_a_gap']}
• Theorem 2
• Remark 3
• Theorem 4
• Theorem 5: Informal version of Theorem \ref{['thm:recursive_clsutering']}
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10