| Abstract: |
| We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Unlike traditional static or heuristic approaches, we recast the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by variational cost functional rather than predefined statistical shapes, this continuous-time formulation naturally accommodates non-parametric cluster evolution. To analytically validate our general framework, we analyze the specific setting of time-dependent Gaussian mixtures. We prove that the MFG dynamics explicitly recover the trajectories of the classic Expectation-Maximization (EM) algorithm, providing a rigorous generalization that guarantees mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to smooth short-term fluctuations. Numerical experiments confirm the validity of our approach in parametric contexts and pave the way for fully non-parametric clustering applications where classical EM methods are not applicable. |
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