# Learning Flat Latent Manifolds with VAEs

arXiv: https://arxiv.org/abs/2002.04881 · HTML: https://mlsanity.com/paper/2002.04881/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2020.

## Built on by (most cited first, 30 of 33)

- [Computationally-Efficient Neural Image Compression with Shallow Decoders](https://mlsanity.com/paper/2304.06244/) (2023)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Isometric Representation Learning for Disentangled Latent Space of Diffusion Models](https://mlsanity.com/paper/2407.11451/) (2024)
- [Local Disentanglement in Variational Auto-Encoders Using Jacobian L1 Regularization](https://mlsanity.com/paper/2106.02923/) (2021)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Amortized Variational Inference: A Systematic Review](https://mlsanity.com/paper/2209.10888/) (2022)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [On Explicit Curvature Regularization in Deep Generative Models](https://mlsanity.com/paper/2309.10237/) (2023)
- [Data Representations' Study of Latent Image Manifolds](https://mlsanity.com/paper/2305.19730/) (2023)
- [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)
- [Out-of-distribution Detection and Generation using Soft Brownian Offset Sampling and Autoencoders](https://mlsanity.com/paper/2105.02965/) (2021)
- [A Geometric Perspective on Autoencoders](https://mlsanity.com/paper/2309.08247/) (2023)
- [GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductions](https://mlsanity.com/paper/2206.05183/) (2022)
- [Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation](https://mlsanity.com/paper/2507.11789/) (2025)
- [Adversarial robustness of VAEs through the lens of local geometry](https://mlsanity.com/paper/2208.03923/) (2022)
- [Disentangled Representation Learning with the Gromov-Monge Gap](https://mlsanity.com/paper/2407.07829/) (2024)
- [Latent Space Characterization of Autoencoder Variants](https://mlsanity.com/paper/2412.04755/) (2024)
- [Flat Latent Manifolds for Human-machine Co-creation of Music](https://mlsanity.com/paper/2202.12243/) (2022)
- [Understanding Variational Autoencoders with Intrinsic Dimension and Information Imbalance](https://mlsanity.com/paper/2411.01978/) (2024)
- [Curvature Enhanced Data Augmentation for Regression](https://mlsanity.com/paper/2506.06853/) (2025)
- [Bi-Lipschitz Autoencoder With Injectivity Guarantee](https://mlsanity.com/paper/2604.06701/) (2026)
- [Discriminating Against Unrealistic Interpolations in Generative Adversarial Networks](https://mlsanity.com/paper/2203.01035/) (2022)
- [Sequential Model for Predicting Patient Adherence in Subcutaneous Immunotherapy for Allergic Rhinitis](https://mlsanity.com/paper/2401.11447/) (2024)
- [Modularity aided consistent attributed graph clustering via coarsening](https://mlsanity.com/paper/2407.07128/) (2024)
- [Investigating Image Manifolds of 3D Objects: Learning, Shape Analysis, and Comparisons](https://mlsanity.com/paper/2503.06773/) (2025)
- [Expressive Score-Based Priors for Distribution Matching with Geometry-Preserving Regularization](https://mlsanity.com/paper/2506.14607/) (2025)
- [Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoders](https://mlsanity.com/paper/2508.20413/) (2025)
- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)

## Abstract (arXiv)

Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact latent space. Prevalent is the use of the Euclidean metric, which has the drawback of ignoring information about similarity of data stored in the decoder, as captured by the framework of Riemannian geometry. We propose an extension to the framework of variational auto-encoders allows learning flat latent manifolds, where the Euclidean metric is a proxy for the similarity between data points. This is achieved by defining the latent space as a Riemannian manifold and by regularising the metric tensor to be a scaled identity matrix. Additionally, we replace the compact prior typically used in variational auto-encoders with a recently presented, more expressive hierarchical one---and formulate the learning problem as a constrained optimisation problem. We evaluate our method on a range of data-sets, including a video-tracking benchmark, where the performance of our unsupervised approach nears that of state-of-the-art supervised approaches, while retaining the computational efficiency of straight-line-based approaches.