Maximum likelihood estimation (MLE) of latent variable models is often recast as an optimization problem over the extended space of parameters and probability distributions. For example, the Expectation Maximization (EM) algorithm can be interpreted as coordinate descent applied to a suitable free energy functional over this space. Recently, this perspective has been combined with insights from optimal transport and Wasserstein gradient flows to develop particle-based algorithms applicable to wider classes of models than standard EM. Drawing inspiration from prior works which interpret `momentum-enriched' optimisation algorithms as discretizations of ordinary differential equations, we propose an analogous dynamical systems-inspired approach to minimizing the free energy functional over the extended space of parameters and probability distributions. The result is a dynamic system that blends elements of Nesterov's Accelerated Gradient method, the underdamped Langevin diffusion, and particle methods. Under suitable assumptions, we establish quantitative convergence of the proposed system to the unique minimiser of the functional in continuous time. We then propose a numerical discretization of this system which enables its application to parameter estimation in latent variable models. Through numerical experiments, we demonstrate that the resulting algorithm converges faster than existing methods and compares favourably with other (approximate) MLE algorithms.

翻译：隐变量模型的最大似然估计通常被重新表述为参数与概率分布扩展空间上的优化问题。例如，期望最大化算法可解释为该空间上合适自由能泛函的坐标下降法。近期，这一视角与最优传输和Wasserstein梯度流理论相结合，发展出适用于比标准EM算法更广泛模型类别的粒子算法。受前人将"含动量"优化算法解释为常微分方程离散化的研究启发，我们提出一种基于动力系统的类比方法，用于在参数与概率分布扩展空间上最小化自由能泛函。由此产生的动力系统融合了Nesterov加速梯度法、欠阻尼Langevin扩散和粒子方法的元素。在适当假设下，我们建立了该系统在连续时间内向泛函唯一极小值点的定量收敛性。随后提出该系统的数值离散化方案，使其能够应用于隐变量模型中的参数估计。数值实验表明，该算法比现有方法收敛更快，且性能优于其他（近似）最大似然估计算法。