Convergent and Anti-diffusive Properties of Mean-Shift Method

TL;DR

This paper develops an analytical PDE-based framework to study the convergence and stability of mean-shift clustering algorithms, revealing their intrinsic instability and proposing supervised modifications for improved convergence.

Contribution

It introduces a novel PDE-based analysis of mean-shift algorithms, demonstrating their inherent instability and suggesting supervised approaches for better convergence.

Findings

Unsupervised mean-shift algorithms are intrinsically unstable.

Correct convergence requires transforming data into a multivariate normal distribution.

Supervised mechanisms can enhance the stability and convergence of mean-shift algorithms.

Abstract

An analytic framework based on partial differential equations is derived for certain dynamic clustering methods. The proposed mathematical framework is based on the application of the conservation law in physics to characterize successive transformations of the underlying probability density function. It is then applied to analyze the convergence and stability of mean shift type of dynamic clustering algorithms. Theoretical analysis shows that un-supervised mean-shift type of algorithm is intrinsically unstable. It is proved that the only possibility of a correct convergence for unsupervised mean shift type of algorithm is to transform the original probability density into a multivariate normal distribution with no dependence struture. Our analytical results suggest that a more stable and convergent mean shift algorithm might be achieved by adopting a judiciously chosen supervision mechanism.