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Cartan moving frames and the data manifolds

Eliot Tron, Rita Fioresi, Nicolas Couellan, Stéphane Puechmorel

TL;DR

Using 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, 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.

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.

Cartan moving frames and the data manifolds

TL;DR

Using 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, 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.

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.
Paper Structure (11 sections, 20 theorems, 67 equations, 6 figures, 5 tables)

This paper contains 11 sections, 20 theorems, 67 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cartan moving frames
  4. The Curvature of the Data Leaves
  5. Computation of the curvature forms
  6. Experiments
  7. Conclusion
  8. Cartan moving frames
  9. An auxiliary lemma
  10. Further computations of the curvature forms
  11. More experiments

Key Result

Theorem 2.4

The data information matrix $D(x)$ is positive semidefinite, moreover with $\perp$ taken w.r.t. the Euclidean scalar product on $T\mathcal{X}$. Hence, the rank of $D$ is bounded by $C-1$, with $C$ the number of classes.

Figures (6)

  • Figure 1: Couples of input point $x$ (above) and the corresponding matrix $\hat{D}(x) = \left( g^D_x\left( e_a,e_b \right) \right)_{a,b=1,\ldots,C}$ (below) on MNIST.
  • Figure 2: Couples of input point $x$ (above) and the corresponding matrix $\hat{D}(x) = \left( g^D_x\left( e_a,e_b \right) \right)_{a,b=1,\ldots,C}$ (below) on CIFAR10.
  • Figure 3: Frenet-Serret Frames
  • Figure 4: Soldering form on a sphere.
  • Figure 5: Couples of input point $x$ (above) and the corresponding matrix $\hat{D}(x) = \left( g^D_x\left( e_a,e_b \right) \right)_{a,b=1,\ldots,C}$ (below) on MNIST used for Table \ref{['tab:MNIST-experiment-seed-42']} numbered from left to right and from top to bottom.
  • ...and 1 more figures

Theorems & Definitions (45)

  • Definition 2.1
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • ...and 35 more