# Trainable and Explainable Simplicial Map Neural Networks

arXiv: https://arxiv.org/abs/2306.00010 · HTML: https://mlsanity.com/paper/2306.00010/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 18 references that are in the map.
- First posted on arXiv 29 May 2023.

## Builds on (most cited first, 6 of 6)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Interpretable machine learning: definitions, methods, and applications](https://mlsanity.com/paper/1901.04592/) (2019)
- [Understanding Neural Networks via Feature Visualization: A survey](https://mlsanity.com/paper/1904.08939/) (2019)
- [Understanding Instance-based Interpretability of Variational Auto-Encoders](https://mlsanity.com/paper/2105.14203/) (2021)
- [On the approximation by single hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/1708.06219/) (2017)
- [Topology-based Representative Datasets to Reduce Neural Network Training Resources](https://mlsanity.com/paper/1903.08519/) (2019)

## Built on by (most cited first, 2 of 2)

- [Topological Data Analysis Applications in Natural Language Processing: A Survey](https://mlsanity.com/paper/2411.10298/) (2024)
- [Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximation](https://mlsanity.com/paper/2509.06694/) (2025)

## Abstract (arXiv)

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.