K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence

• URL: http://arxiv.org/abs/2603.04625v1
• Date: Wed, 04 Mar 2026 21:41:50 GMT
• Title: K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence
• Authors: Felipe de Jesus Felix Arredondo, Alejandro Ucan-Puc, Carlos Astengo Noguez,
• Abstract summary: This work establishes a rigorous variational and gradient-based equivalence between the classical K-Means algorithm and differentiable Basis Radial Function neural networks.<n>We show that the RBF objective $$-converges to the K-Means solution as the temperature parameter $$ vanishes.<n>We also propose the integration of Ent-max-1.5, which ensures stable convergence while preserving the underlying Voronoi partition structure.
• Score: 41.99844472131922
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work establishes a rigorous variational and gradient-based equivalence between the classical K-Means algorithm and differentiable Radial Basis Function (RBF) neural networks with smooth responsibilities. By reparameterizing the K-Means objective and embedding its distortion functional into a smooth weighted loss, we prove that the RBF objective $Γ$-converges to the K-Means solution as the temperature parameter $σ$ vanishes. We further demonstrate that the gradient-based updates of the RBF centers recover the exact K-Means centroid update rule and induce identical training trajectories in the limit. To address the numerical instability of the Softmax transformation in the low-temperature regime, we propose the integration of Entmax-1.5, which ensures stable polynomial convergence while preserving the underlying Voronoi partition structure. These results bridge the conceptual gap between discrete partitioning and continuous optimization, enabling K-Means to be embedded directly into deep learning architectures for the joint optimization of representations and clusters. Empirical validation across diverse synthetic geometries confirms a monotone collapse of soft RBF centroids toward K-Means fixed points, providing a unified framework for end-to-end differentiable clustering.

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