# Variational Autoencoders with Riemannian Brownian Motion Priors

arXiv: https://arxiv.org/abs/2002.05227 · HTML: https://mlsanity.com/paper/2002.05227/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2020.

## Built on by (most cited first, 30 of 34)

- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Pythae: Unifying Generative Autoencoders in Python -- A Benchmarking Use Case](https://mlsanity.com/paper/2206.08309/) (2022)
- [A Geometric Perspective on Variational Autoencoders](https://mlsanity.com/paper/2209.07370/) (2022)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [EditVAE: Unsupervised Part-Aware Controllable 3D Point Cloud Shape Generation](https://mlsanity.com/paper/2110.06679/) (2021)
- [Preventing Manifold Intrusion with Locality: Local Mixup](https://mlsanity.com/paper/2201.04368/) (2022)
- [Uncertainty Estimation Using Riemannian Model Dynamics for Offline Reinforcement Learning](https://mlsanity.com/paper/2102.11327/) (2021)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [Counterfactual Explanations via Riemannian Latent Space Traversal](https://mlsanity.com/paper/2411.02259/) (2024)
- [GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductions](https://mlsanity.com/paper/2206.05183/) (2022)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Decoder ensembling for learned latent geometries](https://mlsanity.com/paper/2408.07507/) (2024)
- [Quadratic Matrix Factorization with Applications to Manifold Learning](https://mlsanity.com/paper/2301.12965/) (2023)
- [Learning Identity-Preserving Transformations on Data Manifolds](https://mlsanity.com/paper/2106.12096/) (2021)
- [On the Generative Utility of Cyclic Conditionals](https://mlsanity.com/paper/2106.15962/) (2021)
- [On the Latent Holes of VAEs for Text Generation](https://mlsanity.com/paper/2110.03318/) (2021)
- [Variational autoencoder with weighted samples for high-dimensional non-parametric adaptive importance sampling](https://mlsanity.com/paper/2310.09194/) (2023)
- [Riemannian Generative Decoder](https://mlsanity.com/paper/2506.19133/) (2025)
- [Visualizing Riemannian data with Rie-SNE](https://mlsanity.com/paper/2203.09253/) (2022)
- [Riemannian-Manifold Steering: Geometry-Aware Generative Autoencoders for Label-Free Steering](https://mlsanity.com/paper/2605.24942/) (2026)
- [Constructing VAE Latent Spaces with Prescribed Topology](https://mlsanity.com/paper/2606.07058/) (2026)
- [Boltzmann Tuning of Generative Models](https://mlsanity.com/paper/2104.05252/) (2021)
- [Variational Inference for Longitudinal Data Using Normalizing Flows](https://mlsanity.com/paper/2303.14220/) (2023)
- [Illumination and Shadows in Head Rotation: experiments with Denoising Diffusion Models](https://mlsanity.com/paper/2308.06057/) (2023)
- [Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds](https://mlsanity.com/paper/2310.06157/) (2023)
- [Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoders](https://mlsanity.com/paper/2508.20413/) (2025)
- [Privacy-Utility Trade-off in Data Publication: A Bilevel Optimization Framework with Curvature-Guided Perturbation](https://mlsanity.com/paper/2509.02048/) (2025)
- [PepCompass: Navigating peptide embedding spaces using Riemannian Geometry](https://mlsanity.com/paper/2510.01988/) (2025)

## Abstract (arXiv)

Variational Autoencoders (VAEs) represent the given data in a low-dimensional latent space, which is generally assumed to be Euclidean. This assumption naturally leads to the common choice of a standard Gaussian prior over continuous latent variables. Recent work has, however, shown that this prior has a detrimental effect on model capacity, leading to subpar performance. We propose that the Euclidean assumption lies at the heart of this failure mode. To counter this, we assume a Riemannian structure over the latent space, which constitutes a more principled geometric view of the latent codes, and replace the standard Gaussian prior with a Riemannian Brownian motion prior. We propose an efficient inference scheme that does not rely on the unknown normalizing factor of this prior. Finally, we demonstrate that this prior significantly increases model capacity using only one additional scalar parameter.