# HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference

## TL;DR

HINT introduces a hierarchical invertible neural network architecture that enhances expressiveness for density estimation and Bayesian inference by recursively applying coupling transformations, enabling efficient sampling and posterior computation.

## Contribution

The paper proposes a recursive hierarchical coupling scheme for invertible neural networks, improving their expressiveness and applicability in density estimation and Bayesian inference.

## Key findings

- Effective density estimation on standard datasets.
- Accurate Bayesian inference for 2D shape data.
- Enhanced sample visualization across dimensions.

## Abstract

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.