基于去噪扩散的精确全局马尔可夫链蒙特卡洛（MCMC）
Exact Global MCMC with Denoising Diffusion
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中文总结 AI 辅助
本研究提出去噪扩散蒙特卡洛（DDMC）方法，将去噪扩散模型与MALA结合，为复杂高维目标密度提供高接受率的全局MCMC建议，初步验证了标准扩散训练缩放行为可迁移至高维未归一化密度的精确采样。
中文摘要 AI 辅助
本研究表明，通过标准去噪损失训练得到的扩散模型，可为复杂高维目标密度提供有效的全局马尔可夫链蒙特卡洛（MCMC）建议分布。该方法的动机源于以下观察：依次应用正向和反向扩散过程可定义一个马尔可夫链，其目标平稳分布对应于针对目标分布样本训练的理想去噪器。对于任意去噪器，通过应用包含离散时间SDE近似的正向和反向路径密度的接受比的Metropolis-Hastings步骤，可使该观察结果变得精确。因此，我们建议在局部收敛的MALA样本上训练去噪扩散模型，以学习全局MCMC建议分布。我们将基于全局去噪器的路径采样器与局部MALA采样器的组合称为去噪扩散蒙特卡洛（DDMC）。实验表明，DDMC在多种复杂目标密度上可提供具有高接受率的全局建议分布。我们的结果提供了初步证据，表明标准扩散训练的既定缩放行为可直接迁移到高维未归一化密度的精确采样中。
英文摘要
This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.
