# The Riemannian Geometry of Deep Generative Models

arXiv: https://arxiv.org/abs/1711.08014 · HTML: https://mlsanity.com/paper/1711.08014/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2017.

## Built on by (most cited first, 30 of 82)

- [Manifold Preserving Guided Diffusion](https://mlsanity.com/paper/2311.16424/) (2023)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Diffeomorphic Counterfactuals with Generative Models](https://mlsanity.com/paper/2206.05075/) (2022)
- [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)
- [From Bricks to Bridges: Product of Invariances to Enhance Latent Space Communication](https://mlsanity.com/paper/2310.01211/) (2023)
- [Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution](https://mlsanity.com/paper/2405.09800/) (2024)
- [Which Models have Perceptually-Aligned Gradients? An Explanation via Off-Manifold Robustness](https://mlsanity.com/paper/2305.19101/) (2023)
- [The Manifold Hypothesis for Gradient-Based Explanations](https://mlsanity.com/paper/2206.07387/) (2022)
- [SRIF: Semantic Shape Registration Empowered by Diffusion-based Image Morphing and Flow Estimation](https://mlsanity.com/paper/2409.11682/) (2024)
- [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)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [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)
- [Unsupervised Disentanglement of Linear-Encoded Facial Semantics](https://mlsanity.com/paper/2103.16605/) (2021)
- [A unified model for continuous conditional video prediction](https://mlsanity.com/paper/2210.05810/) (2022)
- [Adaptive Feature Interpolation for Low-Shot Image Generation](https://mlsanity.com/paper/2112.02450/) (2021)
- [Autoencoding Low-Resolution MRI for Semantically Smooth Interpolation of Anisotropic MRI](https://mlsanity.com/paper/2202.09258/) (2022)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Latent Spectral Regularization for Continual Learning](https://mlsanity.com/paper/2301.03345/) (2023)
- [Unsupervised Discovery of Interpretable Directions in h-space of Pre-trained Diffusion Models](https://mlsanity.com/paper/2310.09912/) (2023)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [MMP++: Motion Manifold Primitives with Parametric Curve Models](https://mlsanity.com/paper/2310.17072/) (2023)
- [Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning](https://mlsanity.com/paper/2410.13792/) (2024)
- [The Weight Gram Matrix Captures Sequential Feature Linearization in Deep Networks](https://mlsanity.com/paper/2605.06258/) (2026)

## Abstract (arXiv)

Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. However, further investigation into this phenomenon is warranted, to identify if there are other architectures or datasets where curvature plays a more prominent role. We believe that exploring the Riemannian geometry of deep generative models, using the tools developed in this paper, will be an important step in understanding the high-dimensional, nonlinear spaces these models learn.