# Fast Approximate Geodesics for Deep Generative Models

arXiv: https://arxiv.org/abs/1812.08284 · HTML: https://mlsanity.com/paper/1812.08284/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2018.

## Built on by (most cited first, 13 of 13)

- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [Uncertainty Estimation Using Riemannian Model Dynamics for Offline Reinforcement Learning](https://mlsanity.com/paper/2102.11327/) (2021)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [Embedding-Aligned Language Models](https://mlsanity.com/paper/2406.00024/) (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)
- [Boundary-Aware Uncertainty for Feature Attribution Explainers](https://mlsanity.com/paper/2210.02419/) (2022)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [Using the Path of Least Resistance to Explain Deep Networks](https://mlsanity.com/paper/2502.12108/) (2025)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Flat Latent Manifolds for Human-machine Co-creation of Music](https://mlsanity.com/paper/2202.12243/) (2022)
- [Discriminating Against Unrealistic Interpolations in Generative Adversarial Networks](https://mlsanity.com/paper/2203.01035/) (2022)
- [Self-Balancing, Memory Efficient, Dynamic Metric Space Data Maintenance, for Rapid Multi-Kernel Estimation](https://mlsanity.com/paper/2504.18003/) (2025)

## Abstract (arXiv)

The length of the geodesic between two data points along a Riemannian manifold, induced by a deep generative model, yields a principled measure of similarity. Current approaches are limited to low-dimensional latent spaces, due to the computational complexity of solving a non-convex optimisation problem. We propose finding shortest paths in a finite graph of samples from the aggregate approximate posterior, that can be solved exactly, at greatly reduced runtime, and without a notable loss in quality. Our approach, therefore, is hence applicable to high-dimensional problems, e.g., in the visual domain. We validate our approach empirically on a series of experiments using variational autoencoders applied to image data, including the Chair, FashionMNIST, and human movement data sets.