[Paper Review] Clustering Partially Observed Graphs via Convex Optimization

Yudong Chen, Ali Jalali+2 more|arXiv (Cornell University)|Apr 25, 2011

Sparse and Compressive Sensing Techniques44 references97 citations

TL;DR

This paper proposes a convex optimization framework for clustering partially observed unweighted graphs by minimizing disagreements—missing edges within clusters and present edges across clusters. It reduces the problem to low-rank and sparse matrix decomposition from partial observations, achieving exact clustering recovery under mild conditions on cluster size, edge density, and observation probability, with optimality up to logarithmic factors in the stochastic block model.

ABSTRACT

This paper considers the problem of clustering a partially observed unweighted graph---i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"---i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.

Motivation & Objective

• To address the challenge of clustering graphs where edge information is only partially observed—some edges are known to exist, some to be absent, and others unobserved.
• To develop a clustering method that minimizes disagreements without requiring prior knowledge of the number of clusters.
• To provide a provably correct algorithm that either returns the optimal clustering or detects failure, avoiding suboptimal solutions.
• To establish theoretical conditions under which exact recovery is possible in the planted partition/stochastic block model with partial observations.
• To characterize the tradeoff between observation probability and edge density gap for successful clustering.

Proposed method

• The method models the observed adjacency matrix as the sum of a low-rank matrix (representing ideal cluster structure) and a sparse matrix (representing disagreements).
• It uses convex optimization to recover the low-rank and sparse components via nuclear norm and l1-norm minimization on partially observed entries.
• The algorithm solves a matrix decomposition problem: minimize ‖L‖_* + λ‖S‖_1 subject to P_Ω(A) = P_Ω(L + S), where L is low-rank and S is sparse.
• The solution is derived from a novel reduction of disagreement minimization to robust matrix decomposition under partial observations.
• Theoretical analysis relies on concentration inequalities and matrix perturbation theory to bound the infinity norm of residual terms.
• Recovery is guaranteed when the observation probability and edge density gap satisfy certain thresholds relative to cluster size and graph size.

Experimental results

Research questions

• RQ1 Can we achieve exact clustering of partially observed graphs using a convex optimization framework without pre-specifying the number of clusters?
• RQ2 What are the sufficient conditions on observation probability, edge density gap, and cluster size for successful recovery in the stochastic block model?
• RQ3 How does the proposed method compare to existing approaches in terms of theoretical guarantees and performance under partial observations?
• RQ4 To what extent does the method achieve optimal recovery up to logarithmic factors in the worst-case scenario?
• RQ5 Can the disagreement minimization objective be effectively solved via convex relaxation with provable correctness?

Key findings

• The proposed algorithm guarantees exact clustering recovery when the observation probability and edge density gap satisfy p₀ ≥ C log n / K_min, where K_min is the minimum cluster size.
• For constant-sized clusters of equal size, the method achieves optimality up to logarithmic factors in the recovery threshold.
• Theoretical analysis shows that the algorithm succeeds with high probability when the observation probability p₀ is sufficiently large relative to the edge density gap and cluster size.
• The method outperforms general matrix decomposition results by providing stronger guarantees tailored to the clustering problem.
• The recovery threshold scales as O(log n / K_min), which matches known information-theoretic limits up to logarithmic factors.
• Failure occurs only when the data do not satisfy the theoretical conditions, ensuring no suboptimal solutions are returned.