A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel
This addresses sampling and generative modeling problems in statistics and machine learning, representing an incremental improvement with adaptive bandwidth selection.
The authors tackled the problem of approximating target distributions by proposing EVI-MMD, a deterministic sampling method that minimizes Maximum Mean Discrepancy using an energetic variational inference framework with adaptive kernel bandwidth. They showed it outperforms existing methods for both fully specified density functions and two-sample problems.
We propose a novel deterministic sampling method to approximate a target distribution $ρ^*$ 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.