# A Note on the Convergence of the Gaussian Mean Shift Algorithm

## TL;DR

This paper analyzes the convergence of the Gaussian Mean Shift algorithm by framing it as an MM algorithm, extending previous results to include non-isolated stationary points.

## Contribution

It demonstrates that the Gaussian Mean Shift algorithm can be viewed as an MM algorithm, allowing new convergence insights, especially for non-isolated stationary points.

## Key findings

- MS fixed points are stationary points of KDE
- Convergence can be analyzed via MM framework
- Extends convergence results to non-isolated stationary points

## Abstract

Mean shift (MS) algorithms are popular methods for mode finding in pattern analysis. Each MS algorithm can be phrased as a fixed-point iteration scheme, which operates on a kernel density estimate (KDE) based on some data. The ability of an MS algorithm to obtain the modes of its KDE depends on whether or not the fixed-point scheme converges. The convergence of MS algorithms have recently been proved under some general conditions via first principle arguments. We complement the recent proofs by demonstrating that the MS algorithm operating on a Gaussian KDE can be viewed as an MM (minorization-maximization) algorithm, and thus permits the application of convergence techniques for such constructions. For the Gaussian case, we extend upon the previously results by showing that the fixed-points of the MS algorithm are all stationary points of the KDE in cases where the stationary points may not necessarily be isolated.