# Explorations in Homeomorphic Variational Auto-Encoding

arXiv: https://arxiv.org/abs/1807.04689 · HTML: https://mlsanity.com/paper/1807.04689/

## Key facts

- Cited by 65 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2018.

## Built on by (most cited first, 30 of 65)

- [GTA: A Geometry-Aware Attention Mechanism for Multi-View Transformers](https://mlsanity.com/paper/2310.10375/) (2023)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Self-supervised learning of Split Invariant Equivariant representations](https://mlsanity.com/paper/2302.10283/) (2023)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [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)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Image to Sphere: Learning Equivariant Features for Efficient Pose Prediction](https://mlsanity.com/paper/2302.13926/) (2023)
- [Equivariant Representation Learning via Class-Pose Decomposition](https://mlsanity.com/paper/2207.03116/) (2022)
- [One-shot Imitation Learning via Interaction Warping](https://mlsanity.com/paper/2306.12392/) (2023)
- [Unsupervised Learning of Equivariant Structure from Sequences](https://mlsanity.com/paper/2210.05972/) (2022)
- [Synthetic Data Supervised Salient Object Detection](https://mlsanity.com/paper/2210.13835/) (2022)
- [Addressing the Topological Defects of Disentanglement via Distributed Operators](https://mlsanity.com/paper/2102.05623/) (2021)
- [Neural Fourier Transform: A General Approach to Equivariant Representation Learning](https://mlsanity.com/paper/2305.18484/) (2023)
- [On Incorporating Inductive Biases into VAEs](https://mlsanity.com/paper/2106.13746/) (2021)
- [Continuous normalizing flows on manifolds](https://mlsanity.com/paper/2104.14959/) (2021)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [Approximate Equivariance in Reinforcement Learning](https://mlsanity.com/paper/2411.04225/) (2024)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductions](https://mlsanity.com/paper/2206.05183/) (2022)
- [Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distribution](https://mlsanity.com/paper/2308.14666/) (2023)
- [Pose-Aware Self-Supervised Learning with Viewpoint Trajectory Regularization](https://mlsanity.com/paper/2403.14973/) (2024)
- [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://mlsanity.com/paper/2307.03704/) (2023)
- [Learning Rotation Invariant Features for Cryogenic Electron Microscopy Image Reconstruction](https://mlsanity.com/paper/2101.03549/) (2021)
- [FONDUE: an algorithm to find the optimal dimensionality of the latent representations of variational autoencoders](https://mlsanity.com/paper/2209.12806/) (2022)

## Abstract (arXiv)

The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds with the Gaussian prior assumption typically made in Variational Auto-Encoders (VAEs), because the density of a Gaussian concentrates near a blob-like manifold. In this paper we investigate the use of manifold-valued latent variables. Specifically, we focus on the important case of continuously differentiable symmetry groups (Lie groups), such as the group of 3D rotations operatornameSO(3). We show how a VAE with operatornameSO(3)-valued latent variables can be constructed, by extending the reparameterization trick to compact connected Lie groups. Our experiments show that choosing manifold-valued latent variables that match the topology of the latent data manifold, is crucial to preserve the topological structure and learn a well-behaved latent space.