证明了特定核函数下均值漂移算法的收敛性，并发现大带宽下聚类效果依赖于核函数类型。
Convergence and clustering analysis for Mean Shift with radially symmetric, positive definite kernels
- 使用径向对称正定核函数，证明大带宽下算法在任意维度收敛
- 实验表明高斯核在大带宽时难以实现准确聚类，其他核可例外
- 为均值漂移算法的理论分析提供新视角，适合研究聚类算法的学者
均值漂移（Mean Shift, MS）是一种非参数、基于密度的迭代算法，在聚类和图像分割中广泛应用。其模式估计序列在一般情况下的收敛性尚未有严格证明。本文证明：当带宽足够大时，对于任意维度及任意径向对称且严格正定的核函数，均值漂移算法的收敛性可被保证。尽管该结果在带宽下限上较文献[YT]更受限，但所用核类不被[YT]覆盖，且证明方法不同。此外，理论与实验均表明，对于高斯核，大带宽下通常无法实现精确聚类；而对于其他径向对称、严格正定核函数，仍有可能实现。此结果深化了对均值漂移行为的理解，尤其揭示了核函数选择在大带宽场景中的关键作用。
原文摘要 · Abstract (English)
The mean shift (MS) is a non-parametric, density-based, iterative algorithm with prominent usage in clustering and image segmentation. A rigorous proof for the convergence of its mode estimate sequence in full generality remains unknown. In this paper, we show that for\textit{ sufficiently large bandwidth} convergence is guaranteed in any dimension with \textit{any radially symmetric and strictly positive definite kernels}. Although the author acknowledges that our result is partially more restrictive than that of \cite{YT} due to the lower limit of the bandwidth, our kernel class is not covered by the kernel class in \cite{YT}, and the proof technique is different. Moreover, we show theoretically and experimentally that while for Gaussian kernel, accurate clustering at \textit{large bandwidths} is generally impossible, it may still be possible for other radially symmetric, strictly positive definite kernels.
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