  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.
