二次K-medoids图聚类的常模优化

中文摘要 AI 辅助

图聚类旨在将图的顶点划分为组，使得组内连接密集而组间连接稀疏。本文从常模（CM）优化的角度研究图聚类的二次K-medoids模型。通过将二元可行集识别为离散常模集，我们建立了二次K-medoids模型与常模优化之间的联系。我们不直接求解原始二元问题，而是考虑其可行集的凸包，并引入负平方范数惩罚以促进极端点解，从而在一个简单凸集上得到极端点追踪模型。该凸集上的高效投影使得基于投影梯度的优化能够用于大规模medoid选择。基于此模型，我们提出了一种两阶段聚类方法。在第一阶段，使用惩罚连续投影梯度求解器近似求解带惩罚的medoid选择问题。在第二阶段，通过统一的两级分配规则将选定的medoids转换为图划分，其中Jaccard距离作为主要准则，必要时使用最短路径距离作为次要准则。在合成PPM和SBM图以及真实世界图数据上的数值实验证明了我们方法的有效性和计算效率。实验还表明，medoid选择问题中较小的目标值不一定导致更准确的最终图划分，这凸显了在分配后同时评估优化性能和聚类质量的重要性。

英文摘要

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.

发表机构

• School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）

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