This paper argues for the use of normalizing flows - a way of building up new probability distributions by applying multiple sets of invertible transformations to existing distributions - as a way of building more flexible variational inference models. 

The central premise of a variational autoencoder is that of learning an approximation to the posterior distribution of latent variables - p(z|x) - and parameterizing that distribution according to values produced by a neural network. In typical practice, this has meant that VAEs are limited in terms of the complexity of latent variable distributions they can encode, since using an analytically specified distribution tends to limit you to simpler distributional shapes - Gaussians, uniform, and the like. Normalizing flows are here proposed as a way to allow for the model to learn more complex forms of posterior distribution. 

Normalizing flows work off of a fairly simple intuition: if you take samples from a distribution p(x), and then apply a function f(x) to each x in that sample, you can calculate the expected value of your new distribution f(x) by calculating the expectation of f(x) under the old distribution p(x). That is to say: 
https://i.imgur.com/NStm7zN.png
This mathematical transformation has a pretty delightful name - The Law of the Unconscious Statistician - that came from the fact that so many statisticians just treated this identity as a definitional fact, rather than something actually in need of proving (I very much fall into this bucket as well). The implication of this is that if you apply many transformations in sequence to the draws from some simple distribution, you can work with that distribution without explicitly knowing its analytical formulation, just by being able to evaluate - and, importantly - invert the function. The ability to invert the function is key, because of the way you calculate the derivative: by taking the inverse of the determinant of the derivative of your function f(z) with respect to z. (Note here that q(z) is the original distribution you sampled under, and q’(z) is the implicit density you’re trying to estimate, after your function has been applied). 

https://i.imgur.com/8LmA0rc.png

Combining these ideas together: a variational flow autoencoder works by having an encoder network define the parameters of a simple distribution (Gaussian or Uniform), and then running the samples from that distribution through a series of k transformation layers. This final transformed density over z is then given to the decoder to work with. Some important limitations are in place here, the most salient of which is that in order to calculate derivatives, you have to be able to calculate the determinant of the derivative of a given transformation. Due to this constraint, the paper only tests a few transformations where this is easy to calculate analytically - the planar transformation and radial transformation. If you think about transformations of density functions as fundamentally stretching or compressing regions of density, the planar transformation works by stretching along an axis perpendicular to some parametrically defined plane, and the radial transformation works by stretching outward in a radial way around some parametrically defined point. Even though these transformations are individually fairly simple, when combined, they can give you a lot more flexibility in distributional space than a simple Gaussian or Uniform could. 

https://i.imgur.com/Xf8HgHl.png