K-Means作为径向基函数网络：变分与梯度等价
K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence
AI总结：
本文通过变分和梯度方法证明K-Means与RBF网络等价，提出Entmax-1.5解决低温稳定性问题，实现K-Means在深度学习中的直接嵌入与联合优化。
摘要：
本文建立了经典的K-Means算法与可微的径向基函数（RBF）神经网络之间严格的变分和梯度等价关系。通过重新参数化K-Means目标函数，并将其形变函数嵌入到平滑加权损失中，我们证明了当温度参数σ趋近于零时，RBF目标函数Γ收敛到K-Means解。我们进一步展示了RBF中心的梯度更新能够恢复精确的K-Means质心更新规则，并在极限情况下诱导相同的训练轨迹。为了解决低温范围内Softmax变换的数值不稳定性问题，我们提出整合Entmax-1.5，这确保了稳定的多项式收敛性同时保持底层Voronoi划分结构。这些结果弥合了离散划分与连续优化之间的概念鸿沟，使K-Means能够直接嵌入到深度学习架构中，用于表示和聚类的联合优化。在多样化的合成几何上进行的实证验证证实了软RBF质心单调地坍缩到K-Means固定点，提供了一个统一的端到端可微聚类框架。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
