K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence
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| Format: | Preprint |
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2026
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| author | Arredondo, Felipe de Jesus Felix Ucan-Puc, Alejandro Noguez, Carlos Astengo |
| author_facet | Arredondo, Felipe de Jesus Felix Ucan-Puc, Alejandro Noguez, Carlos Astengo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_04625 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence Arredondo, Felipe de Jesus Felix Ucan-Puc, Alejandro Noguez, Carlos Astengo Machine Learning Statistics Theory 62H30 62H30 62H30 62H30, 62F12, 49J45, 68T05 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. |
| title | K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence |
| topic | Machine Learning Statistics Theory 62H30 62H30 62H30 62H30, 62F12, 49J45, 68T05 |
| url | https://arxiv.org/abs/2603.04625 |