Constant-Modulus Optimization for Quadratic K-Medoids Graph Clustering
Abstract
Graph clustering aims to partition the vertices of a graph into groups with dense intra-cluster connections and sparse inter-cluster connections. In this paper, we study a quadratic K-medoids formulation for graph clustering from the perspective of CM optimization. By identifying the binary feasible set as a discrete CM set, we establish a connection between the quadratic K-medoids model and CM optimization. Rather than solving the original binary problem directly, we consider the convex hull of its feasible set and introduce a negative squared-norm penalty to promote extreme-point solutions, resulting in an extreme point pursuit model over a simple convex set. The efficient projection onto this convex set enables the use of projected-gradient-based optimization for large-scale medoid selection. Based on this formulation, we propose a two-stage clustering method. In Stage~1, a penalty-continuation projected gradient solver is used to approximately solve the penalized medoid-selection problem. In Stage~2, the selected medoids are converted into a graph partition through a unified two-level assignment rule, where Jaccard distance is used as the primary criterion and shortest-path distance is used as a secondary criterion when needed. Numerical experiments on synthetic PPM and SBM graphs, together with real-world graph data, demonstrate the effectiveness and computational efficiency of our method. The experiments also show that a smaller objective value in the medoid-selection problem does not necessarily lead to a more accurate final graph partition, highlighting the importance of evaluating both optimization performance and clustering quality after assignment.
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BibTeX
@misc{zheng2026constantmodulus,
title = {{Constant-Modulus Optimization for Quadratic K-Medoids Graph Clustering}},
author = {Yutong Zheng and Qingna Li and Wenshun Teng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01404},
eprint = {2610.01404},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01404},
}
RIS
TY - UNPB
TI - Constant-Modulus Optimization for Quadratic K-Medoids Graph Clustering
AU - Yutong Zheng
AU - Qingna Li
AU - Wenshun Teng
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01404
UR - https://arxiv.org/abs/2610.01404
AB - Graph clustering aims to partition the vertices of a graph into groups with dense intra-cluster connections and sparse inter-cluster connections. In this paper, we study a quadratic K-medoids formulation for graph clustering from the perspective of CM optimization. By identifying the binary feasible set as a discrete CM set, we establish a connection between the quadratic K-medoids model and CM optimization. Rather than solving the original binary problem directly, we consider the convex hull of its feasible set and introduce a negative squared-norm penalty to promote extreme-point solutions, resulting in an extreme point pursuit model over a simple convex set. The efficient projection onto this convex set enables the use of projected-gradient-based optimization for large-scale medoid selection. Based on this formulation, we propose a two-stage clustering method. In Stage~1, a penalty-continuation projected gradient solver is used to approximately solve the penalized medoid-selection problem. In Stage~2, the selected medoids are converted into a graph partition through a unified two-level assignment rule, where Jaccard distance is used as the primary criterion and shortest-path distance is used as a secondary criterion when needed. Numerical experiments on synthetic PPM and SBM graphs, together with real-world graph data, demonstrate the effectiveness and computational efficiency of our method. The experiments also show that a smaller objective value in the medoid-selection problem does not necessarily lead to a more accurate final graph partition, highlighting the importance of evaluating both optimization performance and clustering quality after assignment.
ER -