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Trainable and Explainable Simplicial Map Neural Networks

Eduardo Paluzo-Hidalgo, Miguel A. Gutiérrez-Naranjo, Rocio Gonzalez-Diaz

TL;DR

This paper proposes an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere, and introduces the explainability capacity of SMNNs and an effective implementation.

Abstract

Simplicial map neural networks (SMNNs) are topology-based neural networks with interesting properties such as universal approximation ability and robustness to adversarial examples under appropriate conditions. However, SMNNs present some bottlenecks for their possible application in high-dimensional datasets. First, SMNNs have precomputed fixed weight and no SMNN training process has been defined so far, so they lack generalization ability. Second, SMNNs require the construction of a convex polytope surrounding the input dataset. In this paper, we overcome these issues by proposing an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere. In addition, the explainability capacity of SMNNs and an effective implementation are also newly introduced in this paper.

Trainable and Explainable Simplicial Map Neural Networks

TL;DR

This paper proposes an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere, and introduces the explainability capacity of SMNNs and an effective implementation.

Abstract

Simplicial map neural networks (SMNNs) are topology-based neural networks with interesting properties such as universal approximation ability and robustness to adversarial examples under appropriate conditions. However, SMNNs present some bottlenecks for their possible application in high-dimensional datasets. First, SMNNs have precomputed fixed weight and no SMNN training process has been defined so far, so they lack generalization ability. Second, SMNNs require the construction of a convex polytope surrounding the input dataset. In this paper, we overcome these issues by proposing an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere. In addition, the explainability capacity of SMNNs and an effective implementation are also newly introduced in this paper.
Paper Structure (13 sections, 4 theorems, 34 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 4 theorems, 34 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Simplicial complexes
  4. Simplicial maps
  5. Simplicial maps for classification tasks
  6. Simplicial map neural networks
  7. The unknown boundary and the function $\varphi_{ U}$
  8. Training SMNNs
  9. Explainability
  10. Discussion and Limitations
  11. Experiments
  12. Conclusions
  13. Aknowledgments

Key Result

Lemma 1

Let $x\in B^n$. Then,

Figures (8)

  • Figure 1: On the left, two triangles that do not intersect in a common face (an edge or a vertex). On the right, the geometric representation $|K|$ of a pure 2-simplicial complex $K$ composed of three maximal $2$-simplices (the triangles $\sigma^1$, $\sigma^2$ and $\sigma^3$). The edge $\mu^2$ is a common face of $\sigma^2$ and $\sigma^3$. The edge $\mu^1$ is a face of $\sigma^1$.
  • Figure 2: Illustration of a simplicial map for a classification task.
  • Figure 3: An example of the point $w^x$ computed from $x$ and the $(n-1)$-simplex $\mu=\langle u^{1},u^{2}\rangle\in \Gamma$ such that $x\in|\sigma|$ for $\sigma=\langle w^x,u^{1},u^{2}\rangle$.
  • Figure 4: The relative positions of the vertices $\tilde{v}^i$ for $i\in [\![1,5]\!]$ and the points $\tilde{x}^1$ and $\tilde{x}^2$ of Example \ref{['example']}.
  • Figure 5: Two-dimensional synthetic dataset with points divided into two classes: blue and yellow. Triangle-shaped points belong to the test set and square-shaped points belong to the support set $U$. The diamond-shaped point is the vertex on the hypersphere (the blue circumference) used to classify the triangle-shaped point $v$ (surrounded by a small red circumference) outside the triangulation.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 1
  • Example 1
  • Lemma 1: Continuity
  • Lemma 2: Consistence
  • Lemma 3: $\mathcal{L}$-optimum simplicial map
  • Theorem 1