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A Mean Field Games Perspective on Evolutionary Clustering

Alessio Basti, Fabio Camilli, Adriano Festa

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.

A Mean Field Games Perspective on Evolutionary 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.

Paper Structure

This paper contains 11 sections, 5 theorems, 82 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Non-Parametric Mean Field Games Framework
  3. Fokker--Planck Equation and the Gaussian Case
  4. $1^{st}$ model: maximizing the instantaneous log-likelihood
  5. $2^{nd}$ model: maximizing a time average of the log-likelihood
  6. Numerical approximation and examples
  7. Computational approach and discretization
  8. Test 1: A 1D example
  9. Test 2: A 2D example
  10. Conclusions
  11. Appendix

Key Result

Proposition 3.1

Let $m$ in emmfg_mix be a mixture with $(m_k,\alpha_k)$, $k=1,\dots,K$, given by the system emMFG. Then, for each time $t$, $m(\cdot,t)$ is a Gaussian mixture with where

Figures (8)

  • Figure 1: Static baseline: EM applied independently at each time step ($t=0,\dots,5$). The lack of temporal coupling produces inconsistent labels and abrupt shape changes.
  • Figure 2: Instantaneous model (Section 2.1): the FP evolution \ref{['emMFG']} with drift from \ref{['coeff_eq_1model']} without explicit time averaging.
  • Figure 3: Asymmetric relaxation with $\tau=0.5$ (cf. \ref{['eq:asym-relax']}--\ref{['eq:drift-asym']}). The causal EM stabilizes parameters while remaining responsive to genuine shifts.
  • Figure 4: Symmetric relaxation with $\tau=0.5$ (cf. \ref{['eq:sym-relax']}). Bidirectional smoothing produces the most stable paths, with a small latency in tracking sharp transitions.
  • Figure 5: Comparison of the evolution of the centroids at various moments. The asymmetric relaxed (magenta) and symmetric relaxed (red) models exhibit a clear smoothing effect relative to the instantaneous model, which is computed independently at each time instant. Both smoothing approaches yield more stable trajectories; however, the asymmetric relaxed model introduces a latency in capturing the primary transitions.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • Remark 3.7
  • Proposition 3.8
  • Remark 3.9
  • ...and 5 more