Clustering as approximation by constrained projectors: Theory and guarantees

Angshul Majumdar

doi:10.1016/j.sigpro.2026.110641

Clustering as approximation by constrained projectors: Theory and guarantees

2026-04-12

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Angshul Majumdar *

DOI：10.1016/j.sigpro.2026.110641

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This paper presents a unified projector-based framework for analysing a broad class of clustering methods, including k-means, fuzzy c-means (FCM), kernel k-means, kernel FCM, and spectral clustering, through structured low-rank approximations of a signal-derived matrix. Specifically, we show that these methods can be written within the common optimisation template minB∈C∥M−MPB∥F2, where the choice of constraint set C determines the clustering family and the associated projector structure (orthogonal or oblique). This formulation makes explicit which algebraic connections are classical (e.g., kernel and spectral projection viewpoints) and which aspects are unified here under a single constrained optimisation perspective. Within this framework, we establish theoretical results on (i) the geometry of the projector search space, including a geodesic characterisation for orthogonal projector families, (ii) perturbation bounds that quantify stability under matrix noise, and (iii) exact recovery conditions under ideal block-model assumptions. The analysis also identifies conditions under which hard, fuzzy, kernel, and spectral formulations share the same optimal subspace, and explains how small inter-cluster leakage leads to controlled deviations. The resulting framework provides a coherent theory-first foundation for comparing and interpreting clustering methods through structured projectors.

Keywords:

clustering

constrained projectors

structured low-rank approximation

spectral clustering

k-means

Clustering as approximation by constrained projectors: Theory and guarantees

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