Transport maps can ease the sampling of distributions with non-trivial geometries by transforming them into distributions that are easier to handle. The potential of this approach has risen with the development of Normalizing Flows (NF) which are maps parameterized with deep neural networks trained to push a reference distribution towards a target. NF-enhanced samplers recently proposed blend (Markov chain) Monte Carlo methods with either (i) proposal draws from the flow or (ii) a flow-based reparametrization. In both cases, the quality of the learned transport conditions performance. The present work clarifies for the first time the relative strengths and weaknesses of these two approaches. Our study concludes that multimodal targets can be reliably handled with flow-based proposals up to moderately high dimensions. In contrast, methods relying on reparametrization struggle with multimodality but are more robust otherwise in high-dimensional settings and under poor training. To further illustrate the influence of target-proposal adequacy, we also derive a new quantitative bound for the mixing time of the Independent Metropolis-Hastings sampler.

翻译：传输映射能够通过将具有复杂几何结构的分布转化为更易处理的分布，从而简化采样过程。随着正则化流（Normalizing Flows，NF）的发展——即通过深度神经网络参数化、训练将参考分布推向目标分布的映射——该方法的应用潜力日益凸显。近期提出的基于NF增强的采样器将（马尔可夫链）蒙特卡洛方法与以下两种策略相结合：（i）从流模型中生成提议样本，或（ii）基于流模型的重参数化。在这两种情况下，学习到的传输映射质量直接影响模型性能。本研究首次明确阐明了这两种方法的相对优势与局限。我们的分析表明：对于中等至高维度的多模态目标分布，基于流模型的提议方法能够可靠地处理；相比之下，依赖重参数化的方法在多模态场景中表现不佳，但在高维设置及训练不充分的情况下更具鲁棒性。为进一步说明目标分布与提议分布匹配性的影响，我们还推导了独立Metropolis-Hastings采样器混合时间的新定量界。