T-ARC：基于分布鲁棒随机块模型的拓扑感知随机聚类
T-ARC: Topology-Aware Randomized Clustering via Distributionally Robust Stochastic Block Models
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中文总结 AI 辅助
T-ARC通过将拓扑信息嵌入K-means目标并利用分布鲁棒随机块模型，纠正几何偏差，在非凸和交错簇上取得最高准确率且更稳定。
中文摘要 AI 辅助
本文提出了一种新的聚类方法，即T-ARC（拓扑感知随机聚类），通过将拓扑信息直接嵌入优化目标中来纠正K-means的几何偏差。基于数据具有由潜在图建模的隐藏结构这一假设，其思想是通过标准K-means数据保真项与图割惩罚项之间的相互作用来揭示这一信息，图割惩罚项阻止与数据连通性结构不一致的聚类分配。为了使这种耦合易于处理，潜在图被建模为随机块模型（SBM）的随机实现，其标量参数在分布鲁棒优化（DRO）框架内进行优化，从而产生闭式近端更新。SBM和DRO均由基于零维持续同调（$H_0$）的持续性相似矩阵提供信息，该矩阵将数据的多尺度连通性结构转化为成对拓扑先验。整体优化通过块坐标下降进行；通过全局Lyapunov泛函建立收敛性：确定性块满足单调下降，而随机图更新满足期望下降，因此期望能量收敛。在具有非凸几何的合成数据集和Fashion-MNIST的随机子集上的实验表明，T-ARC恢复了K-means失败的潜在拓扑结构，在弯曲和交错簇上实现了最高准确率，同时在真实数据上保持竞争力且明显比K-means更稳定。
英文摘要
In this work, we introduce a new clustering method, namely T-ARC (Topology-Aware Randomized Clustering), that corrects the geometric bias of K-means by embedding topological information directly into the optimization objective. Building on the assumption that the data admits an underlying hidden structure modeled via a latent graph, the idea is to uncover this information through the interplay between the standard K-means data-fidelity term and a graph-cut penalty, which discourages cluster assignments inconsistent with the connectivity structure of the data. To render this coupling tractable, the latent graph is modeled as a random realization from a Stochastic Block Model (SBM), whose scalar parameter is optimized within a Distributionally Robust Optimization (DRO) framework, yielding a closed-form proximal update. Both SBM and DRO are informed by a persistence-based similarity matrix derived from zero-dimensional persistent homology ($H_0$), which translates the multiscale connectivity structure of the data into a pairwise topological prior. The overall optimization proceeds via Block Coordinate Descent; convergence is established through a global Lyapunov functional: the deterministic blocks satisfy monotonic descent, while the stochastic graph update satisfies descent in expectation, so that the expected energy converges. Experiments on synthetic datasets with non-convex geometries and on random subsets of Fashion-MNIST show that T-ARC recovers latent topological structures where K-means fails, achieving the highest accuracy on curved and interleaved clusters while remaining competitive, and markedly more stable than K-means, on real data.
发表机构
- University of Bari Aldo Moro（巴里阿尔多莫罗大学）
- University of Southampton（南安普敦大学）
- National Research Council (CNR)（国家研究委员会）
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