  We introduce manifold-learning flows (M-flows), a new class of generative
models that simultaneously learn the data manifold as well as a tractable
probability density on that manifold. Combining aspects of normalizing flows,
GANs, autoencoders, and energy-based models, they have the potential to
represent datasets with a manifold structure more faithfully and provide
handles on dimensionality reduction, denoising, and out-of-distribution
detection. We argue why such models should not be trained by maximum likelihood
alone and present a new training algorithm that separates manifold and density
updates. In a range of experiments we demonstrate how M-flows learn the data
manifold and allow for better inference than standard flows in the ambient data
space.
