Clustering Partially Observed Graphs via Convex Optimization (1104.4803v4)

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.

• The paper establishes conditions under which convex optimization achieves optimal clustering in partially observed graphs.
• It transforms clustering into a low-rank and sparse matrix recovery problem, outperforming traditional spectral methods in noisy environments.
• Empirical results confirm the method's scalability and effectiveness, eliminating the need to predefine the number of clusters.

Analyzing the Efficacy of Convex Optimization in Clustering Partially Observed Graphs

In the paper of clustering partially observed graphs, the paper titled "Clustering Partially Observed Graphs via Convex Optimization" by Chen et al. employs a sophisticated approach utilizing convex optimization techniques. The authors focus on clustering nodes of a graph, which are partially observed, aiming to minimize classification errors, termed "disagreements."

The primary goal of the research is to assign nodes into disjoint clusters such that intra-cluster connectivity is high while inter-cluster connectivity remains sparse. A defining aspect of the proposed approach is the use of convex optimization to find the clustering that minimizes the total number of disagreements—these disagreements being identified as existing edges between nodes in different clusters or missing edges within the same cluster.

The approach transforms the problem into the recovery of an unknown low-rank matrix and an unknown sparse matrix from their partially observed sum. The method is evaluated mainly in the context of the well-known Planted Partition or Stochastic Block Model. This reduction allows the application of convex relaxation techniques, such as matrix splitting, leveraging the mathematical properties of sparse and low-rank decompositions.

Key Results

• Optimality of Convex Optimization: The authors establish sufficient conditions under which their convex optimization algorithm guarantees successful clustering, specifically for the planted partition model with partial observations. Their theoretical guarantees are strong, even when observation probabilities and the density gap, $1-2\tau$, are low.
• Comparison with Existing Techniques: When evaluated against traditional spectral clustering methods, especially under scenarios where entries are imputed or randomly filled, their method demonstrates superior sensitiveness to network sparsity and noise.
• Numerical Experimentation: Empirical results provided in the paper align closely with the theoretical predictions, underscoring the robustness of the proposed method under different levels of observation probability $p_0$ and graph density parameters.
• Scalability and Efficiency: A significant advantage of their method lies in the non-requirement to specify the number of clusters, $k$, which is often a prerequisite or an loose end of most spectral or $k$-means based approaches.

Implications and Future Directions

The implications of this approach are significant in numerous practical scenarios where graph data may be incomplete, such as social network analysis, biological networks, or VLSI design automation. The proposed method has the potential for broad applications wherein obtaining complete graph information is either impractical or expensive.

Theoretically, the authors have provided pivotal insights into the complexity of clustering with missing data, providing a compelling case for further exploration of convex optimization techniques in robust graph-based learning methods. Further research might extend these ideas beyond strictly unweighted graphs to more diverse graph types such as weighted or directed networks, potentially improving clustering methodologies in various interdisciplinary domains.

Overall, this paper contributes a rigorous examination and a novel solution to clustering in partially observed graphs, enriching both theoretical understanding and practical applications in the field of graph analytics.