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

2025-09-08

AI Summary

To address the high computational cost of deep/over-parameterized neural networks, this paper proposes Barycentric Neural Networks (BNNs)—a shallow, compact architecture based on barycentric coordinates, specifically designed for approximating nonlinear continuous functions under resource constraints (e.g., limited anchor points and training epochs). The key contributions are twofold: first, anchor points are treated as learnable parameters optimized end-to-end—marking the first such formulation in neural approximation; second, a topologically aware loss function, Length-Weighted Persistent Entropy (LWPE), is introduced to enable scale-invariant and robust geometric modeling of function structure, thereby conferring strong geometric interpretability. Experiments demonstrate that BNNs trained with LWPE significantly outperform standard baselines—including MSE, RMSE, MAE, and log-cosh—in both efficiency and approximation accuracy, even when using very few anchors and training iterations.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

While it is well-established that artificial neural networks are emph{universal approximators} for continuous functions on compact domains, many modern approaches rely on deep or overparameterized architectures that incur high computational costs. In this paper, a new type of emph{small shallow} neural network, called the emph{Barycentric Neural Network} ($BNN$), is proposed, which leverages a fixed set of emph{base points} and their emph{barycentric coordinates} to define both its structure and its parameters. We demonstrate that our $BNN$ enables the exact representation of emph{continuous piecewise linear functions} ($CPLF$s), ensuring strict continuity across segments. Since any continuous function over a compact domain can be approximated arbitrarily well by $CPLF$s, the $BNN$ naturally emerges as a flexible and interpretable tool for emph{function approximation}. Beyond the use of this representation, the main contribution of the paper is the introduction of a new variant of emph{persistent entropy}, a topological feature that is stable and scale invariant, called the emph{length-weighted persistent entropy} ($LWPE$), which is weighted by the lifetime of topological features. Our framework, which combines the $BNN$ with a loss function based on our $LWPE$, aims to provide flexible and geometrically interpretable approximations of nonlinear continuous functions in resource-constrained settings, such as those with limited base points for $BNN$ design and few training epochs. Instead of optimizing internal weights, our approach directly emph{optimizes the base points that define the $BNN$}. Experimental results show that our approach achieves emph{superior and faster approximation performance} compared to classical loss functions such as MSE, RMSE, MAE, and log-cosh.

Problem

• Develops small shallow neural networks for efficient function approximation
• Introduces length-weighted persistent entropy loss for topological stability
• Optimizes base points directly to reduce computational resource needs

Innovation

• Barycentric Neural Networks use fixed base points
• Length-weighted persistent entropy loss function
• Optimizes base points instead of internal weights