# Geometrically Enriched Latent Spaces

arXiv: https://arxiv.org/abs/2008.00565 · HTML: https://mlsanity.com/paper/2008.00565/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Aug 2020.

## Built on by (most cited first, 30 of 45)

- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Unsupervised Discovery of Semantic Latent Directions in Diffusion Models](https://mlsanity.com/paper/2302.12469/) (2023)
- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Attribution-based Explanations that Provide Recourse Cannot be Robust](https://mlsanity.com/paper/2205.15834/) (2022)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [The Information Geometry of Softmax: Probing and Steering](https://mlsanity.com/paper/2602.15293/) (2026)
- [Uncertainty Estimation Using Riemannian Model Dynamics for Offline Reinforcement Learning](https://mlsanity.com/paper/2102.11327/) (2021)
- [GeomCA: Geometric Evaluation of Data Representations](https://mlsanity.com/paper/2105.12486/) (2021)
- [Embedding-Aligned Language Models](https://mlsanity.com/paper/2406.00024/) (2024)
- [Counterfactual Explanations via Riemannian Latent Space Traversal](https://mlsanity.com/paper/2411.02259/) (2024)
- [A Geometric Perspective on Autoencoders](https://mlsanity.com/paper/2309.08247/) (2023)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Pullback Flow Matching on Data Manifolds](https://mlsanity.com/paper/2410.04543/) (2024)
- [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)
- [Unsupervised Discovery of Interpretable Directions in h-space of Pre-trained Diffusion Models](https://mlsanity.com/paper/2310.09912/) (2023)
- [Decoder ensembling for learned latent geometries](https://mlsanity.com/paper/2408.07507/) (2024)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [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)
- [Generalized Flow Matching for Transition Dynamics Modeling](https://mlsanity.com/paper/2410.15128/) (2024)
- [Image Interpolation with Score-based Riemannian Metrics of Diffusion Models](https://mlsanity.com/paper/2504.20288/) (2025)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Deep Curvilinear Editing: Commutative and Nonlinear Image Manipulation for Pretrained Deep Generative Model](https://mlsanity.com/paper/2211.14573/) (2022)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Geometry and Generalization: Eigenvalues as predictors of where a network will fail to generalize](https://mlsanity.com/paper/2107.06386/) (2021)
- [Geodesic Calculus on Implicitly Defined Latent Manifolds](https://mlsanity.com/paper/2510.09468/) (2025)

## Abstract (arXiv)

A common assumption in generative models is that the generator immerses the latent space into a Euclidean ambient space. Instead, we consider the ambient space to be a Riemannian manifold, which allows for encoding domain knowledge through the associated Riemannian metric. Shortest paths can then be defined accordingly in the latent space to both follow the learned manifold and respect the ambient geometry. Through careful design of the ambient metric we can ensure that shortest paths are well-behaved even for deterministic generators that otherwise would exhibit a misleading bias. Experimentally we show that our approach improves interpretability of learned representations both using stochastic and deterministic generators.