A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel

• URL: http://arxiv.org/abs/2111.10722v3
• Date: Tue, 11 Mar 2025 16:09:45 GMT
• Title: A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel
• Authors: Yindong Chen, Yiwei Wang, Lulu Kang, Chun Liu,
• Abstract summary: We propose a novel deterministic sampling method to approximate a target distribution $rho*$ by minimizing the kernel discrepancy.<n>We use the EVI-MMD algorithm to solve two types of sampling problems.
• Score: 5.618322163107168
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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