发表机构
University of Chicago; NSF-Simons AI Institute for the Sky (SkAI Institute); University of Washington（芝加哥大学; NSF-西蒙斯天空人工智能研究所（SkAI研究所）; 华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出梯度引导的密度峰值聚类（GGDPC），通过在最近邻上坡搜索前执行梯度上升步骤，解决了DPC在低密度区域路径不稳定的问题，并在五个标准下建立了其一致性理论。
AI 中文摘要
密度峰值聚类（DPC）将每个观测值连接到其更高密度的最近邻，并将高密度观测值中具有异常大的最近邻上坡位移的观测值识别为聚类中心。然而，从观测值到聚类中心的上坡路径在低密度区域可能不规则且不稳定，这使得聚类分配对局部扰动敏感，并模糊了DPC图的总体几何结构。在本文中，我们引入了梯度引导的密度峰值聚类（GGDPC），它在每次最近邻上坡搜索之前执行一次梯度上升步骤。我们发展了一个稳定性理论，将GGDPC图与总体密度的梯度上升流联系起来。特别地，我们在五个互补标准下建立了GGDPC的一致性：局部模态恢复、调整兰德指数、树状图（聚类树）、路径长度和瀑布度量。这些结果共同为DPC型聚类算法提供了新的统计、几何和拓扑解释。
英文摘要
Density peak clustering (DPC) connects each observation to its nearest neighbor of higher density and identifies cluster centers as high-density observations with unusually large nearest neighbor uphill shifts. The resulting uphill paths from observations to cluster centers, however, can be irregular and unstable in low-density regions, making the clustering assignments sensitive to local perturbations and obscuring the population geometry of the DPC graph. In this paper, we introduce \emph{gradient-guided density peak clustering} (GGDPC), which performs a gradient ascent step before each nearest neighbor uphill search. We develop a stability theory that relates the GGDPC graph to the gradient ascent flow of the population density. In particular, we establish consistency of GGDPC under five complementary criteria: recovery of local modes, adjusted Rand index, dendrogram (cluster tree), path length, and waterfall measure. Together, these results provide new statistical, geometric, and topological interpretations of DPC-type clustering algorithms.
Comments91 pages (27 pages for the main paper), 4 figures, 1 table
