Canonical normalizing flows for manifold learning

Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable invertible transformations. Therefore, once the manifold is properly aligned via a reconstruction loss, the probability density is tractable on the manifold and maximum likelihood can be used to optimize the network parameters. Naturally, the lower-dimensional representation of the data requires an injective-mapping. Recent approaches were able to enforce that the density aligns with the modelled manifold, while efficiently calculating the density volume-change term when embedding to the higher-dimensional space.