  We propose injective generative models called Trumpets that generalize
invertible normalizing flows. The proposed generators progressively increase
dimension from a low-dimensional latent space. We demonstrate that Trumpets can
be trained orders of magnitudes faster than standard flows while yielding
samples of comparable or better quality. They retain many of the advantages of
the standard flows such as training based on maximum likelihood and a fast,
exact inverse of the generator. Since Trumpets are injective and have fast
inverses, they can be effectively used for downstream Bayesian inference. To
wit, we use Trumpet priors for maximum a posteriori estimation in the context
of image reconstruction from compressive measurements, outperforming
competitive baselines in terms of reconstruction quality and speed. We then
propose an efficient method for posterior characterization and uncertainty
quantification with Trumpets by taking advantage of the low-dimensional latent
space.
