Clustering as Approximation by Constrained Projectors: Theory and Guarantees

2026-08-29

AI Summary

本文通过将多种聚类方法统一表达为受约束的低秩投影，建立了一个通用优化框架，并提供了关于这些方法稳定性和恢复保证的理论分析。

Abstract

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.

Problem

• clustering
• structured projectors
• unified theoretical framework
• low-rank projectors
• optimization template

Innovation

• structured low-rank projectors
• unified theoretical framework
• geodesic convexity
• perturbation bounds
• exact recovery guarantees