Manifold lifting: scaling Markov chain Monte Carlo to the vanishing noise regime
针对高信噪比下后验分布集中在低维流形附近导致采样困难的问题，提出流形提升策略，将原采样转化为高维空间中的扩散分布采样，并用约束哈密顿蒙特卡洛方法高效近似推断，数值实验表明采样效率不随目标分布集中而退化。
Abstract Standard Markov chain Monte Carlo methods struggle to explore distributions that concentrate in the neighbourhood of low-dimensional submanifolds. This pathology naturally occurs in Bayesian inference settings when there is a high signal-to-noise ratio in the observational data but the model is inherently over-parametrised or nonidentifiable. In this paper, we propose a strategy that transforms the original sampling problem into the task of exploring a distribution supported on a manifold embedded in a higher-dimensional space; in contrast to the original posterior this lifted distribution remains diffuse in the limit of vanishing observation noise. We employ a constrained Hamiltonian Monte Carlo method, which exploits the geometry of this lifted distribution, to perform efficient approximate inference. We demonstrate in numerical experiments that, contrarily to competing approaches, the sampling efficiency of our proposed methodology does not degenerate as the target distribution to be explored concentrates near low-dimensional submanifolds. Python code reproducing the results is available at https://doi.org/10.5281/zenodo.6551654.