# Latent Space Oddity: on the Curvature of Deep Generative Models

arXiv: https://arxiv.org/abs/1710.11379 · HTML: https://mlsanity.com/paper/1710.11379/

## Key facts

- Cited by 174 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2017.

## Built on by (most cited first, 30 of 174)

- [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)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Norm-guided latent space exploration for text-to-image generation](https://mlsanity.com/paper/2306.08687/) (2023)
- [Qimera: Data-free Quantization with Synthetic Boundary Supporting Samples](https://mlsanity.com/paper/2111.02625/) (2021)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [The Geometry of Deep Generative Image Models and its Applications](https://mlsanity.com/paper/2101.06006/) (2021)
- [Smoothing the Disentangled Latent Style Space for Unsupervised Image-to-Image Translation](https://mlsanity.com/paper/2106.09016/) (2021)
- [Indeterminacy in Generative Models: Characterization and Strong Identifiability](https://mlsanity.com/paper/2206.00801/) (2022)
- [Directional Connectivity-based Segmentation of Medical Images](https://mlsanity.com/paper/2304.00145/) (2023)
- [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)
- [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)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Isometric Representation Learning for Disentangled Latent Space of Diffusion Models](https://mlsanity.com/paper/2407.11451/) (2024)
- [Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution](https://mlsanity.com/paper/2405.09800/) (2024)
- [Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functional](https://mlsanity.com/paper/2504.18506/) (2025)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [Multi-Asset Spot and Option Market Simulation](https://mlsanity.com/paper/2112.06823/) (2021)
- [Latent Traversals in Generative Models as Potential Flows](https://mlsanity.com/paper/2304.12944/) (2023)
- [Data Augmentation with Variational Autoencoders and Manifold Sampling](https://mlsanity.com/paper/2103.13751/) (2021)
- [The Geometry of Neural Nets' Parameter Spaces Under Reparametrization](https://mlsanity.com/paper/2302.07384/) (2023)
- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [DeepBlur: A Simple and Effective Method for Natural Image Obfuscation](https://mlsanity.com/paper/2104.02655/) (2021)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [Lagrangian Manifold Monte Carlo on Monge Patches](https://mlsanity.com/paper/2202.00755/) (2022)
- [Data Representations' Study of Latent Image Manifolds](https://mlsanity.com/paper/2305.19730/) (2023)
- [The Information Geometry of Softmax: Probing and Steering](https://mlsanity.com/paper/2602.15293/) (2026)

## Abstract (arXiv)

Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator imply that the latent space gives a distorted view of the input space. Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and demonstrate that distances and interpolants are significantly improved under this metric. This in turn improves probability distributions, sampling algorithms and clustering in the latent space. Our geometric analysis further reveals that current generators provide poor variance estimates and we propose a new generator architecture with vastly improved variance estimates. Results are demonstrated on convolutional and fully connected variational autoencoders, but the formalism easily generalize to other deep generative models.