# Riemann2: Learning Riemannian Submanifolds from Riemannian Data

arXiv: https://arxiv.org/abs/2503.05540 · HTML: https://mlsanity.com/paper/2503.05540/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 59 references that are in the map.
- First posted on arXiv 7 Mar 2025.

## Builds on (most cited first, 23 of 23)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [NRDF: Neural Riemannian Distance Fields for Learning Articulated Pose Priors](https://mlsanity.com/paper/2403.03122/) (2024)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [What is a meaningful representation of protein sequences?](https://mlsanity.com/paper/2012.02679/) (2020)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [Learning Weighted Submanifolds with Variational Autoencoders and Riemannian Variational Autoencoders](https://mlsanity.com/paper/1911.08147/) (2019)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Uncertainty Estimation Using Riemannian Model Dynamics for Offline Reinforcement Learning](https://mlsanity.com/paper/2102.11327/) (2021)
- [Orientation Probabilistic Movement Primitives on Riemannian Manifolds](https://mlsanity.com/paper/2110.15036/) (2021)
- [Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifolds](https://mlsanity.com/paper/2210.01672/) (2022)
- [Active Learning based on Data Uncertainty and Model Sensitivity](https://mlsanity.com/paper/1808.02026/) (2018)
- [Probabilistic Riemannian submanifold learning with wrapped Gaussian process latent variable models](https://mlsanity.com/paper/1805.09122/) (2018)
- [How Good are Low-Rank Approximations in Gaussian Process Regression?](https://mlsanity.com/paper/2004.01584/) (2020)

## Built on by (most cited first, 6 of 6)

- [Learning Geometry and Topology via Multi-Chart Flows](https://mlsanity.com/paper/2505.24665/) (2025)
- [A probabilistic view on Riemannian machine learning models for SPD matrices](https://mlsanity.com/paper/2505.02402/) (2025)
- [Taxonomy-aware Dynamic Motion Generation on Hyperbolic Manifolds](https://mlsanity.com/paper/2509.21281/) (2025)
- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)
- [Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods](https://mlsanity.com/paper/2609.31107/) (2026)
- [Amortized Structured Stochastic Variational Inference for Gaussian Process Latent Variable Models](https://mlsanity.com/paper/2610.03647/) (2026)

## Abstract (arXiv)

Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches ignore the underlying geometric constraints or fail to provide meaningful metrics in the latent space. To address these limitations, we propose to learn Riemannian latent representations of such geometric data. To do so, we estimate the pullback metric induced by a Wrapped Gaussian Process Latent Variable Model, which explicitly accounts for the data geometry. This enables us to define geometry-aware notions of distance and shortest paths in the latent space, while ensuring that our model only assigns probability mass to the data manifold. This generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.