# Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximation

arXiv: https://arxiv.org/abs/2509.06694 · HTML: https://mlsanity.com/paper/2509.06694/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 8 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks](https://mlsanity.com/paper/1610.09887/) (2016)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Optimal Function Approximation with Relu Neural Networks](https://mlsanity.com/paper/1909.03731/) (2019)
- [A Topological Loss Function: Image Denoising on a Low-Light Dataset](https://mlsanity.com/paper/2208.04573/) (2022)
- [Variational Barycentric Coordinates](https://mlsanity.com/paper/2310.03861/) (2023)
- [Diffeomorphic interpolation for efficient persistence-based topological optimization](https://mlsanity.com/paper/2405.18820/) (2024)
- [On the Convergence of Optimizing Persistent-Homology-Based Losses](https://mlsanity.com/paper/2206.02946/) (2022)
- [Trainable and Explainable Simplicial Map Neural Networks](https://mlsanity.com/paper/2306.00010/) (2023)
- [Safe and Efficient Social Navigation through Explainable Safety Regions Based on Topological Features](https://mlsanity.com/paper/2503.16441/) (2025)

## Abstract (arXiv)

While artificial neural networks are known as universal approximators for continuous functions, many modern approaches rely on overparameterized architectures with high computational cost. In this work, we introduce the Barycentric Neural Network (BNN): a compact shallow architecture that encodes both structure and parameters through a fixed set of base points and their associated barycentric coordinates. We show that the BNN enables the exact representation of continuous piecewise linear functions (CPLFs), ensuring strict continuity across segments. Given that any continuous function on a compact domain can be uniformly approximated by CPLFs, the BNN emerges as a flexible and interpretable tool for function approximation. To enhance geometric fidelity in low-resource scenarios, such as those with few base points to create BNNs or limited training epochs, we propose length-weighted persistent entropy (LWPE): a stable variant of persistent entropy. Our approach integrates the BNN with a loss function based on LWPE to optimize the base points that define the BNN, rather than its internal parameters. Experimental results show that our approach achieves superior and faster approximation performance compared to standard losses (MSE, RMSE, MAE and LogCosh), offering a computationally sustainable alternative for function approximation.