多核谱聚类：逐元素特征向量扰动界与精确恢复

中文摘要 AI 辅助

采用单一带宽的核谱聚类对于呈现多特征成对距离尺度的数据可能效果不佳，这一问题在高维场景中尤为普遍。我们通过聚合不同带宽核的多核公式来解决该问题，带宽被选为成对平方距离的经验分位数，从而在无需先验总体尺度信息的情况下捕捉相关距离尺度。我们在具有异构簇中心和协方差几何结构的通用高维多尺度混合模型下，对所得方法开展严格理论分析。我们构造了经验多核矩阵的分块常数低秩信息近似，并建立了其主导谱分量及关联归一化拉普拉斯矩阵的逐行ℓ₂,∞扰动界。这些界对谱嵌入实现了观测级别的控制，相比传统全局特征空间扰动估计更具信息性。在合适的特征间隙和簇分离条件下，我们证明将近似K-means应用于多核谱嵌入可高概率实现精确恢复。

英文摘要

Kernel spectral clustering with a single bandwidth can be inadequate for data exhibiting multiple characteristic pairwise-distance scales, a problem particularly prevalent in the high-dimensional regime. We address this issue through a multi-kernel formulation that aggregates kernels with different bandwidths. The bandwidths are selected as prescribed empirical quantiles of the pairwise squared distances, thereby capturing the relevant distance scales without requiring prior population-scale information. We develop a rigorous theoretical analysis of the resulting method under a general high-dimensional, multi-scale mixture model with heterogeneous cluster centers and covariance geometries. We construct a blockwise constant, low-rank informative approximation to the empirical multi-kernel matrix and establish row-wise $\ell_{2,\infty}$ perturbation bounds for its leading spectral components, as well as for the associated normalized Laplacian matrix. These bounds yield observation-level control of the spectral embedding, which is more informative than conventional global eigenspace perturbation estimates. Under suitable eigen-gap and cluster-separation conditions, we show that approximate $K$-means applied to the multi-kernel spectral embedding achieves exact recovery with high probability.

发表机构

• Nanyang Technological University（南洋理工大学）
• University of Macau（澳门大学）
• Dalian University of Technology（大连理工大学）

机构由 AI 辅助整理，请以论文原文为准。