作为约束投影器近似的聚类：理论与保证
Clustering as Approximation by Constrained Projectors: Theory and Guarantees
- Indraprastha Institute of Information Technology Delhi（因德拉普拉斯塔信息技术学院（德里））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究提出统一理论框架，将k-means等多种聚类方法表述为结构化低秩投影器，推导测地凸性等理论结果，为理解聚类提供理论优先的基础。
AI中文摘要：
本文提出了一个统一理论框架，表明包括k-means、模糊c-means、核k-means、核FCM和谱聚类在内的广泛聚类方法，均可表示为作用于信号衍生矩阵的结构化低秩投影器。通过将每种方法表述为min_{B∈C} ||M - M P_B||_F^2的实例（其中C为不同约束集），我们建立了一个通用优化模板，阐明了硬、模糊、核诱导及正交投影之间的代数关联。在该框架内，我们推导了非平凡理论结果，包括投影流形上的测地凸性性质、量化矩阵噪声稳定性的扰动界，以及理想块模型条件下的精确恢复保证。分析还进一步解释了不同聚类族何时会坍缩为同一最优子空间，以及在小类间泄漏下偏差如何产生。总体而言，该工作为通过结构化投影器理解聚类提供了连贯的、理论优先的基础。
英文摘要：
This paper develops a unified theoretical framework showing that a broad family of clustering methods, including k-means, fuzzy c-means, kernel k-means, kernel FCM, and spectral clustering, can all be expressed as structured low-rank projectors acting on a signal-derived matrix. By formulating each method as an instance of min over B in C of ||M - M P_B||_F^2, with different constraint sets C, we establish a common optimization template that clarifies the algebraic links among hard, fuzzy, kernel-induced, and orthonormal projections. Within this framework, we derive non-trivial theoretical results, including geodesic convexity properties on the projection manifold, perturbation bounds quantifying stability to matrix noise, and exact recovery guarantees under ideal block-model conditions. The analysis further explains when different clustering families collapse to the same optimal subspace and how deviations arise under small inter-cluster leakage. Overall, the work provides a coherent, theory-first foundation for understanding clustering through structured projectors.
