Diffusion-based supervised learning of generative models for efficient sampling of multimodal distributions
This addresses the challenge of accurate multimodal sampling in Bayesian inference, which is crucial for fields like computational statistics and scientific computing, though it appears to be an incremental improvement over existing methods.
The paper tackles the problem of efficiently sampling from high-dimensional multimodal distributions for Bayesian inference by proposing a hybrid generative model that combines mode identification, domain decomposition, and diffusion-model-assisted training for each mode. The framework effectively handles multimodal distributions with varying shapes in up to 100 dimensions, as demonstrated through numerical examples and an application to Bayesian inverse problems for PDEs.
We propose a hybrid generative model for efficient sampling of high-dimensional, multimodal probability distributions for Bayesian inference. Traditional Monte Carlo methods, such as the Metropolis-Hastings and Langevin Monte Carlo sampling methods, are effective for sampling from single-mode distributions in high-dimensional spaces. However, these methods struggle to produce samples with the correct proportions for each mode in multimodal distributions, especially for distributions with well separated modes. To address the challenges posed by multimodality, we adopt a divide-and-conquer strategy. We start by minimizing the energy function with initial guesses uniformly distributed within the prior domain to identify all the modes of the energy function. Then, we train a classifier to segment the domain corresponding to each mode. After the domain decomposition, we train a diffusion-model-assisted generative model for each identified mode within its support. Once each mode is characterized, we employ bridge sampling to estimate the normalizing constant, allowing us to directly adjust the ratios between the modes. Our numerical examples demonstrate that the proposed framework can effectively handle multimodal distributions with varying mode shapes in up to 100 dimensions. An application to Bayesian inverse problem for partial differential equations is also provided.