A Mean Field Games Perspective on Evolutionary Clustering

TL;DR

This paper introduces a control-theoretic framework for evolutionary clustering using Mean Field Games, enabling flexible, non-parametric cluster evolution analysis with demonstrated stability and mass conservation.

Contribution

It formulates evolutionary clustering as a Mean Field Game, connecting it to classical EM trajectories and providing a novel, variational, continuous-time approach.

Findings

MFG dynamics recover classical EM trajectories

The framework ensures mass conservation in clustering

Numerical experiments show stability and potential for non-parametric clustering

Abstract

We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by a variational cost functional rather than predefined statistical shapes, this continuous-time formulation provides a flexible basis for non-parametric cluster evolution. To validate the framework, we analyze the setting of time-dependent Gaussian mixtures, showing that the MFG dynamics recover the trajectories of the classical Expectation-Maximization (EM) algorithm while ensuring mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to regularize temporal fluctuations. Numerical experiments illustrate the stability of our approach and suggest a path toward more general non-parametric clustering applications where traditional EM methods may face limitations.