# Metrics for Probabilistic Geometries

arXiv: https://arxiv.org/abs/1411.7432 · HTML: https://mlsanity.com/paper/1411.7432/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2014.

## Built on by (most cited first, 28 of 28)

- [Fast Adaptation with Linearized Neural Networks](https://mlsanity.com/paper/2103.01439/) (2021)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [Data Representations' Study of Latent Image Manifolds](https://mlsanity.com/paper/2305.19730/) (2023)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifolds](https://mlsanity.com/paper/2210.01672/) (2022)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics](https://mlsanity.com/paper/2301.06533/) (2023)
- [Decoder ensembling for learned latent geometries](https://mlsanity.com/paper/2408.07507/) (2024)
- [Riemann2: Learning Riemannian Submanifolds from Riemannian Data](https://mlsanity.com/paper/2503.05540/) (2025)
- [Bayesian Quadrature on Riemannian Data Manifolds](https://mlsanity.com/paper/2102.06645/) (2021)
- [Mario Plays on a Manifold: Generating Functional Content in Latent Space through Differential Geometry](https://mlsanity.com/paper/2206.00106/) (2022)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Identifying latent distances with Finslerian geometry](https://mlsanity.com/paper/2212.10010/) (2022)
- [Warped geometric information on the optimisation of Euclidean functions](https://mlsanity.com/paper/2308.08305/) (2023)
- [Identifying Metric Structures of Deep Latent Variable Models](https://mlsanity.com/paper/2502.13757/) (2025)
- [Geometry and Generalization: Eigenvalues as predictors of where a network will fail to generalize](https://mlsanity.com/paper/2107.06386/) (2021)
- [Geodesic Calculus on Implicitly Defined Latent Manifolds](https://mlsanity.com/paper/2510.09468/) (2025)
- [Geometric instability of out of distribution data across autoencoder architecture](https://mlsanity.com/paper/2201.11902/) (2022)
- [On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds](https://mlsanity.com/paper/2410.20850/) (2024)
- [Modeling Barrett's Esophagus Progression using Geometric Variational Autoencoders](https://mlsanity.com/paper/2303.12711/) (2023)
- [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)
- [Aligning the Unseen in Attributed Graphs: Interplay between Graph Geometry and Node Attributes Manifold](https://mlsanity.com/paper/2601.22806/) (2026)
- [Intrinsic-Hybrid Latent Diffusion Models for Generative Modeling on Unknown Manifolds](https://mlsanity.com/paper/2608.04827/) (2026)
- [Bayesian Optimization with Fisher Information Geometry: Gradient Bounds and Trust-Region Methods](https://mlsanity.com/paper/2609.31107/) (2026)
- [Beyond Local Linearity: Scale-Resolved Geometry of Learned Image Encoders](https://mlsanity.com/paper/2609.39115/) (2026)

## Abstract (arXiv)

We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natural metric given by the models. We provide the necessary algorithms to compute expected metric tensors where the distribution over mappings is given by a Gaussian process. We treat the corresponding latent variable model as a Riemannian manifold and we use the expectation of the metric under the Gaussian process prior to define interpolating paths and measure distance between latent points. We show how distances that respect the expected metric lead to more appropriate generation of new data.