Clustering dynamics on graphs: from spectral clustering to mean shift through Fokker-Planck interpolation

TL;DR

This paper develops a unifying framework connecting density-driven and geometry-based clustering algorithms, specifically linking mean shift and spectral clustering via Fokker-Planck equations on graphs, with theoretical insights and numerical validation.

Contribution

It introduces a novel interpolation framework between mean shift and spectral clustering using Fokker-Planck equations, offering new algorithms and theoretical understanding.

Findings

New mean shift algorithms on graphs are proposed.

Theoretical connection between diffusion maps and mean shift dynamics is established.

Numerical examples demonstrate benefits of the interpolation approach.

Abstract

In this work we build a unifying framework to interpolate between density-driven and geometry-based algorithms for data clustering, and specifically, to connect the mean shift algorithm with spectral clustering at discrete and continuum levels. We seek this connection through the introduction of Fokker-Planck equations on data graphs. Besides introducing new forms of mean shift algorithms on graphs, we provide new theoretical insights on the behavior of the family of diffusion maps in the large sample limit as well as provide new connections between diffusion maps and mean shift dynamics on a fixed graph. Several numerical examples illustrate our theoretical findings and highlight the benefits of interpolating density-driven and geometry-based clustering algorithms.