用投影矩阵统一解释多种聚类方法，揭示其内在数学联系。
Clustering as Approximation by Constrained Projectors: Theory and Guarantees
- 将聚类建模为带约束的低秩投影优化问题
- 证明了在理想条件下可精确恢复聚类结构
- 适用于理解聚类稳定性与不同方法间的等价性
本文建立了一个统一的理论框架，表明k-means、模糊c均值、核k均值、核FCM和谱聚类等多种聚类方法均可表示为作用于信号矩阵上的结构化低秩投影算子。通过将每种方法形式化为 min_{B∈C} ||M - MP_B||_F^2，其中约束集C不同，揭示了硬聚类、模糊聚类、核映射及正交投影之间的代数关联。在此框架下，我们推导出非平凡的理论结果，包括投影流形上的测地凸性、噪声扰动下的稳定性界，以及在理想块模型条件下的精确恢复保证。分析还说明了不同聚类族何时收敛至同一最优子空间，并解释了小跨簇泄漏导致偏差的机制。整体工作为从结构投影视角理解聚类提供了理论先导。
原文摘要 · Abstract (English)
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.
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