# A prior-based approximate latent Riemannian metric

arXiv: https://arxiv.org/abs/2103.05290 · HTML: https://mlsanity.com/paper/2103.05290/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 40 references that are in the map.
- First posted on arXiv 9 Mar 2021.

## Builds on (most cited first, 21 of 21)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [Resampled Priors for Variational Autoencoders](https://mlsanity.com/paper/1810.11428/) (2018)
- [Learning Hierarchical Priors in VAEs](https://mlsanity.com/paper/1905.04982/) (2019)
- [Geometrically Enriched Latent Spaces](https://mlsanity.com/paper/2008.00565/) (2020)
- [Unbiased Implicit Variational Inference](https://mlsanity.com/paper/1808.02078/) (2018)
- [Variational Autoencoder with Implicit Optimal Priors](https://mlsanity.com/paper/1809.05284/) (2018)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [Disentangling by Subspace Diffusion](https://mlsanity.com/paper/2006.12982/) (2020)
- [Fast and Robust Shortest Paths on Manifolds Learned from Data](https://mlsanity.com/paper/1901.07229/) (2019)
- [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016)
- [Geodesic Clustering in Deep Generative Models](https://mlsanity.com/paper/1809.04747/) (2018)
- [Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics](https://mlsanity.com/paper/1306.0308/) (2013)
- [Expected path length on random manifolds](https://mlsanity.com/paper/1908.07377/) (2019)

## Built on by (most cited first, 11 of 11)

- [Norm-guided latent space exploration for text-to-image generation](https://mlsanity.com/paper/2306.08687/) (2023)
- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models](https://mlsanity.com/paper/2505.18230/) (2025)
- [Learning Identity-Preserving Transformations on Data Manifolds](https://mlsanity.com/paper/2106.12096/) (2021)
- [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)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [(Deep) Generative Geodesics](https://mlsanity.com/paper/2407.11244/) (2024)
- [Beyond Flatland: A Geometric Take on Matching Methods for Treatment Effect Estimation](https://mlsanity.com/paper/2409.05459/) (2024)

## Abstract (arXiv)

Stochastic generative models enable us to capture the geometric structure of a data manifold lying in a high dimensional space through a Riemannian metric in the latent space. However, its practical use is rather limited mainly due to inevitable complexity. In this work we propose a surrogate conformal Riemannian metric in the latent space of a generative model that is simple, efficient and robust. This metric is based on a learnable prior that we propose to learn using a basic energy-based model. We theoretically analyze the behavior of the proposed metric and show that it is sensible to use in practice. We demonstrate experimentally the efficiency and robustness, as well as the behavior of the new approximate metric. Also, we show the applicability of the proposed methodology for data analysis in the life sciences.