# Geometry-Aware Hamiltonian Variational Auto-Encoder

arXiv: https://arxiv.org/abs/2010.11518 · HTML: https://mlsanity.com/paper/2010.11518/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2020.

## Built on by (most cited first, 10 of 10)

- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [A Geometric Perspective on Variational Autoencoders](https://mlsanity.com/paper/2209.07370/) (2022)
- [Data Augmentation with Variational Autoencoders and Manifold Sampling](https://mlsanity.com/paper/2103.13751/) (2021)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [CLAD: Constrained Latent Action Diffusion for Vision-Language Procedure Planning](https://mlsanity.com/paper/2503.06637/) (2025)
- [Autoencoders for a manifold learning problem with a Jacobian rank constraint](https://mlsanity.com/paper/2306.14194/) (2023)
- [Modeling Barrett's Esophagus Progression using Geometric Variational Autoencoders](https://mlsanity.com/paper/2303.12711/) (2023)
- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)
- [Commutator-Induced Uncertainty in VAEs](https://mlsanity.com/paper/2605.23449/) (2026)

## Abstract (arXiv)

Variational auto-encoders (VAEs) have proven to be a well suited tool for performing dimensionality reduction by extracting latent variables lying in a potentially much smaller dimensional space than the data. Their ability to capture meaningful information from the data can be easily apprehended when considering their capability to generate new realistic samples or perform potentially meaningful interpolations in a much smaller space. However, such generative models may perform poorly when trained on small data sets which are abundant in many real-life fields such as medicine. This may, among others, come from the lack of structure of the latent space, the geometry of which is often under-considered. We thus propose in this paper to see the latent space as a Riemannian manifold endowed with a parametrized metric learned at the same time as the encoder and decoder networks. This metric is then used in what we called the Riemannian Hamiltonian VAE which extends the Hamiltonian VAE introduced by arXiv:1805.11328 to better exploit the underlying geometry of the latent space. We argue that such latent space modelling provides useful information about its underlying structure leading to far more meaningful interpolations, more realistic data-generation and more reliable clustering.