Johann & I released v2 of our paper "Flows for simultaneous manifold learning and density estimation" with more experiments. We dubbed the class of model ℳ-Flows Here you see the flow learning the 2d manifold and the density for the Lorenz attractor! 1/n
arxiv.org/abs/2003.13913
arxiv.org/abs/2003.13913
But there's more. We did our first experiments with images. We used a StyleGAN to produce images that lie on 2-D and 64-D manifolds embedded in a 64x64x3 dimensional data space. Here are some images generated from the model. We have a tractable likelihood for these images! 2/n 
And here we take images, project them onto the manifold (encode/decode like an autoencoder) and look at residuals. (Left/Right are 2-D/64-D manifolds). The ℳ-flow models do very well in learning the manifold (small residuals). We can interpolate between user-supplied images! 3/n
There are also some results using these models for likelihood-free inference for a physics problem. We can learn conditional densities p(x|θ) when x is restricted to / projected onto a manifold x∈ℳ⊂X. That provides a likelihood that we can use to infer p(θ|x) (eg. with MCMC).
