# Geodesic Calculus on Implicitly Defined Latent Manifolds

arXiv: https://arxiv.org/abs/2510.09468 · HTML: https://mlsanity.com/paper/2510.09468/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 52 references that are in the map.
- First posted on arXiv 10 Oct 2025.

## Builds on (most cited first, 25 of 25)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Pose-NDF: Modeling Human Pose Manifolds with Neural Distance Fields](https://mlsanity.com/paper/2207.13807/) (2022)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Metrics for Deep Generative Models](https://mlsanity.com/paper/1711.01204/) (2017)
- [Geometrically Enriched Latent Spaces](https://mlsanity.com/paper/2008.00565/) (2020)
- [Where Do We Stand with Implicit Neural Representations? A Technical and Performance Survey](https://mlsanity.com/paper/2411.03688/) (2024)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [Pythae: Unifying Generative Autoencoders in Python -- A Benchmarking Use Case](https://mlsanity.com/paper/2206.08309/) (2022)
- [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)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [A Review on Riemannian Metric Learning: Closer to You than You Imagine](https://mlsanity.com/paper/2503.05321/) (2025)
- [Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms](https://mlsanity.com/paper/2307.05750/) (2023)
- [Power Weighted Shortest Paths for Clustering Euclidean Data](https://mlsanity.com/paper/1905.13345/) (2019)
- [Learning Distances from Data with Normalizing Flows and Score Matching](https://mlsanity.com/paper/2407.09297/) (2024)
- [Probability Density Geodesics in Image Diffusion Latent Space](https://mlsanity.com/paper/2504.06675/) (2025)
- [Hessian Geometry of Latent Space in Generative Models](https://mlsanity.com/paper/2506.10632/) (2025)
- [Convergent autoencoder approximation of low bending and low distortion manifold embeddings](https://mlsanity.com/paper/2208.10193/) (2022)
- [Identifying latent distances with Finslerian geometry](https://mlsanity.com/paper/2212.10010/) (2022)
- [Geometric optimization using nonlinear rotation-invariant coordinates](https://mlsanity.com/paper/1908.11728/) (2019)

## Built on by (most cited first, 2 of 2)

- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [NeuralSRNF: Neural Square Root Normal Fields for the Statistical Shape Analysis and Generation of Nonrigid 3D and 4D Objects](https://mlsanity.com/paper/2609.27728/) (2026)

## Abstract (arXiv)

Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data.