Paper ID: 2111.10722 • Published Nov 21, 2021
A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel
Yindong Chen, Yiwei Wang, Lulu Kang, Chun Liu
TL;DR
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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.