Convergence and clustering analysis for Mean Shift with radially symmetric, positive definite kernels

TL;DR

This paper proves convergence of the Mean Shift algorithm with large bandwidths for radially symmetric, positive definite kernels and explores the impact of kernel choice on clustering accuracy.

Contribution

It establishes convergence guarantees for Mean Shift with large bandwidths using a broader class of kernels and analyzes kernel effects on clustering performance.

Findings

Convergence is guaranteed for large bandwidths with radially symmetric, positive definite kernels.

Gaussian kernel may not yield accurate clustering at large bandwidths.

Other kernels can achieve better clustering accuracy at large bandwidths.

Abstract

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.