AI 中文总结
针对经典模糊聚类的局限性，提出核模糊关系聚类（KFRC）及基于核Gram矩阵频谱特性的带宽选择算法，通过形式稳定性分析确保其性能稳定，实验证明该框架能恢复传统方法无法解决的复杂结构。
AI 中文摘要
模糊聚类用于通过部分隶属度识别重叠几何聚类结构。经典方法受限于变量重要性相等的假设和对模糊化参数的敏感性。为解决这些问题，我们提出配备通过诱导核Gram矩阵的频谱特性调整的带宽选择算法的核模糊关系聚类（KFRC）。KFRC框架通过可调带宽参数控制数据的几何嵌入，隐式地执行无监督核度量学习。我们进行形式稳定性分析以确定关系聚类崩溃的精确理论条件，从而确保KFRC的稳定性能。我们发现我们的两阶段带宽选择过程适应数据结构，同时积极避免均匀解。此外，这一理论分析导致提出一种新颖的模糊化函数，它比幂模糊化函数具有明显优势。我们在几个合成和公开可用的数据集上进行实验，以证明所提出的框架始终能恢复传统方法无法解决的复杂结构，同时确保纯粹的模糊解。
英文摘要
Fuzzy clustering is used to identify overlapping geometric cluster structures through partial memberships. However, classical methods are limited by the assumption of equal variable importance and by sensitivity to the fuzzifier parameter. These limitations may yield equal cluster membership probabilities, which we refer to as the uniform solution. To address these issues, we propose Kernel Fuzzy Relational Clustering (KFRC) equipped with a bandwidth selection algorithm tuned via the spectral properties of the induced kernel Gram matrix. The KFRC framework implicitly performs unsupervised kernel metric learning by controlling the geometric embedding of the data through adjustable bandwidth parameters. We conduct a formal stability analysis to identify the exact theoretical conditions under which relational clustering collapses, thereby ensuring the stable performance of KFRC. We find that our two-stage bandwidth selection procedure adapts to the data structure while actively avoiding the uniform solution. Furthermore, this theoretical analysis leads to the proposal of a novel fuzzifier function that presents distinct advantages over the power fuzzifier function. We conduct experiments on several synthetic and publicly available data sets to demonstrate that the proposed framework consistently recovers complex structures that traditional methods fail to resolve, while ensuring a purely fuzzy solution.
