We propose a novel deterministic sampling method to approximate a target distribution $\rho^*$ by minimizing the kernel discrepancy, also known as the Maximum Mean Discrepancy (MMD). By employing the general \emph{energetic variational inference} framework (Wang et al., 2021), we convert the problem of minimizing MMD to solving a dynamic ODE system of the particles. We adopt the implicit Euler numerical scheme to solve the ODE systems. This leads to a proximal minimization problem in each iteration of updating the particles, which can be solved by optimization algorithms such as L-BFGS. The proposed method is named EVI-MMD. To overcome the long-existing issue of bandwidth selection of the Gaussian kernel, we propose a novel way to specify the bandwidth dynamically. Through comprehensive numerical studies, we have shown the proposed adaptive bandwidth significantly improves the EVI-MMD. We use the EVI-MMD algorithm to solve two types of sampling problems. In the first type, the target distribution is given by a fully specified density function. The second type is a "two-sample problem", where only training data are available. The EVI-MMD method is used as a generative learning model to generate new samples that follow the same distribution as the training data. With the recommended settings of the tuning parameters, we show that the proposed EVI-MMD method outperforms some existing methods for both types of problems.

翻译：我们提出了一种新颖的确定性采样方法，通过最小化核差异（即最大均值差异，MMD）来逼近目标分布 $\rho^*$。采用通用的**能量变分推断**框架（Wang et al., 2021），我们将最小化MMD的问题转化为求解粒子系统的动态常微分方程（ODE）。我们采用隐式欧拉数值格式求解该ODE系统，这导致在每次粒子更新迭代中产生一个近端最小化问题，该问题可通过L-BFGS等优化算法求解。所提出的方法被命名为EVI-MMD。为克服高斯核带宽选择这一长期存在的问题，我们提出了一种动态指定带宽的新方法。通过全面的数值研究，我们证明了所提出的自适应带宽能显著改进EVI-MMD算法。我们使用EVI-MMD算法求解两类采样问题：第一类中，目标分布由完全指定的密度函数给出；第二类是“双样本问题”，其中仅训练数据可用。EVI-MMD方法作为生成学习模型，用于生成与训练数据同分布的新样本。在推荐的调参设置下，我们证明所提出的EVI-MMD方法在这两类问题上均优于现有的一些方法。