Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SIMAP: A simplicial-map layer for neural networks

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

TL;DR

The paper tackles interpretability in deep learning by introducing the SIMAP layer, a simplicial-map layer that replaces dense final layers with a topology-guided, barycentric-coordinate approach. Unlike prior SMNNs that rely on Delaunay triangulations, SIMAP fixes a maximal simplex and uses barycentric subdivisions plus matrix-based computations to obtain coordinates, enabling scalable, transparent reasoning in $\mathbb{R}^n$. The authors derive the VC-dimension growth under subdivision $VC(\mathcal{N}^k)=((n+1)!)^k\cdot(n+1)$, describe a training procedure that updates weight matrices via gradient descent, and demonstrate the method on synthetic datasets and MNIST in conjunction with a CNN, achieving competitive accuracy. These results suggest SIMAP can enhance explainability without sacrificing performance and can be integrated with standard deep architectures, with code available for reproduction.

Abstract

In this paper, we present SIMAP, a novel layer integrated into deep learning models, aimed at enhancing the interpretability of the output. The SIMAP layer is an enhanced version of Simplicial-Map Neural Networks (SMNNs), an explainable neural network based on support sets and simplicial maps (functions used in topology to transform shapes while preserving their structural connectivity). The novelty of the methodology proposed in this paper is two-fold: Firstly, SIMAP layers work in combination with other deep learning architectures as an interpretable layer substituting classic dense final layers. Secondly, unlike SMNNs, the support set is based on a fixed maximal simplex, the barycentric subdivision being efficiently computed with a matrix-based multiplication algorithm.

SIMAP: A simplicial-map layer for neural networks

TL;DR

The paper tackles interpretability in deep learning by introducing the SIMAP layer, a simplicial-map layer that replaces dense final layers with a topology-guided, barycentric-coordinate approach. Unlike prior SMNNs that rely on Delaunay triangulations, SIMAP fixes a maximal simplex and uses barycentric subdivisions plus matrix-based computations to obtain coordinates, enabling scalable, transparent reasoning in . The authors derive the VC-dimension growth under subdivision , describe a training procedure that updates weight matrices via gradient descent, and demonstrate the method on synthetic datasets and MNIST in conjunction with a CNN, achieving competitive accuracy. These results suggest SIMAP can enhance explainability without sacrificing performance and can be integrated with standard deep architectures, with code available for reproduction.

Abstract

In this paper, we present SIMAP, a novel layer integrated into deep learning models, aimed at enhancing the interpretability of the output. The SIMAP layer is an enhanced version of Simplicial-Map Neural Networks (SMNNs), an explainable neural network based on support sets and simplicial maps (functions used in topology to transform shapes while preserving their structural connectivity). The novelty of the methodology proposed in this paper is two-fold: Firstly, SIMAP layers work in combination with other deep learning architectures as an interpretable layer substituting classic dense final layers. Secondly, unlike SMNNs, the support set is based on a fixed maximal simplex, the barycentric subdivision being efficiently computed with a matrix-based multiplication algorithm.
Paper Structure (12 sections, 6 theorems, 27 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 6 theorems, 27 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorial topology
  4. Neural networks for classification
  5. Simplicial-map layer
  6. Convex polytope and barycentric coordinates
  7. Barycentric coordinates after a barycentric subdivision
  8. Training
  9. Experiments
  10. Experiments with synthetic datasets
  11. Convolutional network in conjunction with a SIMAP layer
  12. Conclusions and future work

Key Result

Lemma 3.1

Let $\mathcal{H}$ be the $n$-dimensional hypercube in $\mathbb{R}^n$ whose corners are the $2^n$ points with coordinates $0$ or $1$. Then, the vertices of an $n$-simplex $\sigma$ satisfying $\mathcal{H}\subset |\sigma|$ are the rows of the following $(n+1)\times n$ matrix

Figures (5)

  • Figure 1: The barycentric subdivision of the simplicial complex described in Example \ref{['example:barycentric']}.
  • Figure 2: All possible dichotomies of a dataset of size $3$ in $\mathbb R^2$ shattered by a line.
  • Figure 3: Dataset of Example \ref{['example:XOR']}. On the left, the input data for the binary classification is shown. On the right, its translation into the simplex $\sigma$ of Lemma \ref{['lemma_tetra']}.
  • Figure 4: Training curves of Example \ref{['example:XOR']}. On the top: the loss function is shown. On the bottom: the accuracy values for the different epochs are shown.
  • Figure 5: From top to bottom, the decision boundaries for Example \ref{['example:XOR']} for the different models with respect to $\sigma$, $\mathop{\mathrm{Sd}}\nolimits \sigma$, and $\mathop{\mathrm{Sd}}\nolimits^2 \sigma$, respectively.

Theorems & Definitions (15)

  • Example 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Example 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 5 more