# Cartan moving frames and the data manifolds

arXiv: https://arxiv.org/abs/2409.12057 · HTML: https://mlsanity.com/paper/2409.12057/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 10 of 10)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Adversarial Examples Are Not Easily Detected: Bypassing Ten Detection Methods](https://mlsanity.com/paper/1705.07263/) (2017)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [The Adversarial Attack and Detection under the Fisher Information Metric](https://mlsanity.com/paper/1810.03806/) (2018)
- [Defending Against Adversarial Attacks by Suppressing the Largest Eigenvalue of Fisher Information Matrix](https://mlsanity.com/paper/1909.06137/) (2019)
- [Model-centric Data Manifold: the Data Through the Eyes of the Model](https://mlsanity.com/paper/2104.13289/) (2021)
- [Inspecting adversarial examples using the Fisher information](https://mlsanity.com/paper/1909.05527/) (2019)
- [Adversarial attacks on neural networks through canonical Riemannian foliations](https://mlsanity.com/paper/2203.00922/) (2022)
- [Two-Stage Metric Learning](https://mlsanity.com/paper/1405.2798/) (2014)
- [Geometry of Singular Foliations and Learning Manifolds in ReLU Networks via the Data Information Matrix](https://mlsanity.com/paper/2409.07412/) (2024)

## Built on by (most cited first, 1 of 1)

- [Multi-Domain Riemannian Graph Gluing for Building Graph Foundation Models](https://mlsanity.com/paper/2603.00618/) (2026)

## Abstract (arXiv)

The purpose of this paper is to employ the language of Cartan moving frames to study the geometry of the data manifolds and its Riemannian structure, via the data information metric and its curvature at data points. Using this framework and through experiments, explanations on the response of a neural network are given by pointing out the output classes that are easily reachable from a given input. This emphasizes how the proposed mathematical relationship between the output of the network and the geometry of its inputs can be exploited as an explainable artificial intelligence tool.