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

arXiv: https://arxiv.org/abs/2603.04625 · HTML: https://mlsanity.com/paper/2603.04625/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 4 Mar 2026.

## Builds on (most cited first, 8 of 8)

- [Deep Clustering for Unsupervised Learning of Visual Features](https://mlsanity.com/paper/1807.05520/) (2018)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Sparse Sequence-to-Sequence Models](https://mlsanity.com/paper/1905.05702/) (2019)
- [Stop using the elbow criterion for k-means and how to choose the number of clusters instead](https://mlsanity.com/paper/2212.12189/) (2022)
- [Deep clustering with concrete k-means](https://mlsanity.com/paper/1910.08031/) (2019)
- [k-means as a variational EM approximation of Gaussian mixture models](https://mlsanity.com/paper/1704.04812/) (2017)
- [Theoretical Analysis of the k-Means Algorithm - A Survey](https://mlsanity.com/paper/1602.08254/) (2016)
- [Rethinking k-means from manifold learning perspective](https://mlsanity.com/paper/2305.07213/) (2023)

## Abstract (arXiv)

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 Gamma-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.