  Many recent invertible neural architectures are based on coupling block
designs where variables are divided in two subsets which serve as inputs of an
easily invertible (usually affine) triangular transformation. While such a
transformation is invertible, its Jacobian is very sparse and thus may lack
expressiveness. This work presents a simple remedy by noting that subdivision
and (affine) coupling can be repeated recursively within the resulting subsets,
leading to an efficiently invertible block with dense, triangular Jacobian. By
formulating our recursive coupling scheme via a hierarchical architecture, HINT
allows sampling from a joint distribution p(y,x) and the corresponding
posterior p(x|y) using a single invertible network. We evaluate our method on
some standard data sets and benchmark its full power for density estimation and
Bayesian inference on a novel data set of 2D shapes in Fourier
parameterization, which enables consistent visualization of samples for
different dimensionalities.
