Cartan moving frames and the data manifolds

TL;DR

This paper introduces a geometric framework using Cartan moving frames to analyze data manifolds and neural network responses, providing a novel approach to explainability in AI through the curvature and structure of data spaces.

Contribution

It applies Cartan moving frames to study data manifold geometry and links this to neural network explainability, a novel integration of differential geometry and AI interpretability.

Findings

Data manifold geometry influences neural network responses.

The curvature of data manifolds can explain class reachability.

The framework enhances interpretability of neural network outputs.

Abstract

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.