On the Sparsifiability of Correlation Clustering: Approximation Guarantees under Edge Sampling
arXiv:2602.13684v1 Announce Type: new Abstract: Correlation Clustering (CC) is a fundamental unsupervised learning primitive whose strongest LP-based approximation guarantees require $\Theta(n^3)$ triangle inequality constraints and are prohibitive at scale. We initiate the study of \emph{sparsification--approximation trade-offs} for CC, asking how much edge information is needed to retain LP-based guarantees. We establish a structural dichotomy between pseudometric and general weighted instances. On the positive side, we prove that the VC dimension of the clustering disagreement class is exactly $n{-}1$, yielding additive $\varepsilon$-coresets of optimal size $\tilde{O}(n/\varepsilon^2)$; that at most $\binom{n}{2}$ triangle inequalities are active at any LP vertex, enabling an exact cutting-plane solver; and that a sparsified variant of LP-PIVOT, which imputes missing LP marginals via triangle inequalities, achieves a robust $\frac{10}{3}$-approximation (up to an additive term cont
arXiv:2602.13684v1 Announce Type: new Abstract: Correlation Clustering (CC) is a fundamental unsupervised learning primitive whose strongest LP-based approximation guarantees require $\Theta(n^3)$ triangle inequality constraints and are prohibitive at scale. We initiate the study of \emph{sparsification--approximation trade-offs} for CC, asking how much edge information is needed to retain LP-based guarantees. We establish a structural dichotomy between pseudometric and general weighted instances. On the positive side, we prove that the VC dimension of the clustering disagreement class is exactly $n{-}1$, yielding additive $\varepsilon$-coresets of optimal size $\tilde{O}(n/\varepsilon^2)$; that at most $\binom{n}{2}$ triangle inequalities are active at any LP vertex, enabling an exact cutting-plane solver; and that a sparsified variant of LP-PIVOT, which imputes missing LP marginals via triangle inequalities, achieves a robust $\frac{10}{3}$-approximation (up to an additive term controlled by an empirically computable imputation-quality statistic $\overline{\Gamma}_w$) once $\tilde{\Theta}(n^{3/2})$ edges are observed, a threshold we prove is sharp. On the negative side, we show via Yao's minimax principle that without pseudometric structure, any algorithm observing $o(n)$ uniformly random edges incurs an unbounded approximation ratio, demonstrating that the pseudometric condition governs not only tractability but also the robustness of CC to incomplete information.
Executive Summary
The article 'On the Sparsifiability of Correlation Clustering: Approximation Guarantees under Edge Sampling' explores the trade-offs between sparsification and approximation in Correlation Clustering (CC), a fundamental unsupervised learning technique. The study reveals a structural dichotomy between pseudometric and general weighted instances, demonstrating that under pseudometric conditions, CC can achieve robust approximation guarantees with significantly fewer edges. The authors establish that the VC dimension of the clustering disagreement class is exactly n-1, enabling efficient coresets and exact cutting-plane solvers. They also introduce a sparsified variant of LP-PIVOT that achieves a 10/3-approximation with O(n^3/2) edges, proving this threshold is sharp. Conversely, without pseudometric structure, any algorithm observing o(n) random edges incurs an unbounded approximation ratio, highlighting the critical role of pseudometric conditions in CC's robustness and tractability.
Key Points
- ▸ Structural dichotomy between pseudometric and general weighted instances in CC.
- ▸ VC dimension of the clustering disagreement class is exactly n-1.
- ▸ Additive ε-coresets of optimal size Õ(n/ε^2) and exact cutting-plane solver enabled by active triangle inequalities.
- ▸ Sparsified LP-PIVOT achieves a robust 10/3-approximation with Õ(n^3/2) edges.
- ▸ Without pseudometric structure, any algorithm observing o(n) random edges incurs an unbounded approximation ratio.
Merits
Theoretical Contributions
The article makes significant theoretical contributions by establishing the VC dimension of the clustering disagreement class and proving the sharpness of the edge threshold for approximation guarantees.
Practical Implications
The findings have practical implications for large-scale CC applications, as they demonstrate that significant sparsification is possible without sacrificing approximation quality under pseudometric conditions.
Robustness Analysis
The study provides a robust analysis of CC's performance under incomplete information, highlighting the importance of pseudometric structure for tractability and robustness.
Demerits
Assumption Dependence
The results are heavily dependent on the pseudometric condition, which may not hold in all real-world scenarios, limiting the generalizability of the findings.
Complexity of Implementation
The proposed methods, such as the sparsified LP-PIVOT, may be complex to implement in practice, requiring further development and validation.
Empirical Validation
While the article provides theoretical guarantees, empirical validation on real-world datasets is limited, which could be a focus for future research.
Expert Commentary
The article presents a rigorous and insightful analysis of the sparsifiability of Correlation Clustering, addressing a critical gap in the understanding of approximation guarantees under edge sampling. The establishment of the VC dimension and the structural dichotomy between pseudometric and general weighted instances are particularly noteworthy. The practical implications of the findings are substantial, as they demonstrate the potential for significant sparsification in CC without compromising approximation quality. However, the dependence on the pseudometric condition and the complexity of implementation are notable limitations. Future research could focus on empirical validation and extending the findings to more general settings. Overall, the article makes a valuable contribution to the field of unsupervised learning and optimization techniques.
Recommendations
- ✓ Further empirical validation of the proposed methods on diverse real-world datasets to assess their practical performance.
- ✓ Exploration of the generalizability of the findings to other unsupervised learning primitives and optimization problems.
- ✓ Development of more accessible and efficient implementations of the proposed algorithms to facilitate their adoption in practical applications.
