Maximum likelihood estimation (MLE) of latent variable models is often recast as the minimization of a free energy functional over an extended space of parameters and probability distributions. This perspective was recently combined with insights from optimal transport to obtain novel particle-based algorithms for fitting latent variable models to data. Drawing inspiration from prior works which interpret `momentum-enriched' optimization algorithms as discretizations of ordinary differential equations, we propose an analogous dynamical-systems-inspired approach to minimizing the free energy functional. The result is a dynamical system that blends elements of Nesterov's Accelerated Gradient method, the underdamped Langevin diffusion, and particle methods. Under suitable assumptions, we prove that the continuous-time system minimizes the functional. By discretizing the system, we obtain a practical algorithm for MLE in latent variable models. The algorithm outperforms existing particle methods in numerical experiments and compares favourably with other MLE algorithms.

翻译：潜变量模型的最大似然估计通常被重构为在参数和概率分布的扩展空间上最小化自由能泛函。这一视角最近与最优传输的洞见相结合，获得了用于将潜变量模型拟合到数据的新型基于粒子的算法。受先前将“动量增强”优化算法解释为常微分方程离散化的工作启发，我们提出了一种类似的、受动力系统启发的自由能泛函最小化方法。其结果是一个融合了Nesterov加速梯度法、欠阻尼朗之万扩散和粒子方法元素的动力系统。在适当假设下，我们证明了该连续时间系统能够最小化该泛函。通过对系统进行离散化，我们得到了一种用于潜变量模型最大似然估计的实用算法。该算法在数值实验中优于现有的粒子方法，并与其他最大似然估计算法相比表现出色。