We consider the problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. In this paper, we build on previous work combining Schr\"odinger bridges and plug & play Langevin samplers. A key bottleneck of these approaches is the exponential dependence of the required training samples on the dimension, $d$, of the ambient state space. We propose a localization strategy which exploits conditional independence of conditional expectation values. Localization thus replaces a single high-dimensional Schr\"odinger bridge problem by $d$ low-dimensional Schr\"odinger bridge problems over the available training samples. In this context, a connection to multi-head self attention transformer architectures is established. As for the original Schr\"odinger bridge sampling approach, the localized sampler is stable and geometric ergodic. The sampler also naturally extends to conditional sampling and to Bayesian inference. We demonstrate the performance of our proposed scheme through experiments on a high-dimensional Gaussian problem, on a temporal stochastic process, and on a stochastic subgrid-scale parametrization conditional sampling problem. We also extend the idea of localization to plug & play Langevin samplers using kernel-based denoising in combination with Tweedie's formula.

翻译：我们考虑从仅能获得足够数量训练样本的未知分布中进行采样的问题。本文基于先前结合薛定谔桥与即插即用朗之万采样器的研究工作展开。这些方法的一个关键瓶颈在于所需训练样本数量与环境状态空间维度$d$呈指数依赖关系。我们提出一种局部化策略，该策略利用条件期望值的条件独立性特性。局部化方法将单一高维薛定谔桥问题转化为$d$个基于可用训练样本的低维薛定谔桥问题。在此背景下，建立了与多头自注意力Transformer架构的关联。与原始薛定谔桥采样方法相同，局部化采样器具有稳定性和几何遍历性。该采样器还能自然扩展到条件采样和贝叶斯推断。我们通过高维高斯问题、时序随机过程以及随机亚网格尺度参数化条件采样问题的实验，验证了所提方案的性能。同时，我们结合基于核的去噪方法与Tweedie公式，将局部化思想扩展到即插即用朗之万采样器。