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

TL;DR

The paper tackles the inefficiency of overparameterized neural networks for function approximation by introducing a shallow, geometry-aware model, the $BNN$, which exactly represents $CPLF$s via fixed base points and barycentric coordinates. It pairs the $BNN$ with a topological loss based on length-weighted persistent entropy ($LWPE$), guiding base-point placement rather than internal weights through the objective $L_{LWPE}$. Empirical results on synthetic and real-world data show faster convergence and higher approximation accuracy under low-resource settings compared to traditional losses (e.g., $MSE$, $RMSE$, $MAE$, LogCosh), aligning with Green AI principles. The framework offers interpretable, scalable function approximation with potential extensions to higher dimensions and hybrid loss strategies.

Abstract

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.

Figures (9)

• Figure 1: The figure shows a simplicial complex with 250 vertices (point-cloud-based function with 250 points) representing the sine function (left), its persistence diagram (middle), barcode, and persistent entropy (right), all from the lower-star filtration.
• Figure 2: Comparison of persistent entropy ($\mathop{\mathrm{PE}}\nolimits$) and length-weighted persistent entropy ($\mathop{\mathrm{LWPE}}\nolimits$) on three subsampled versions of a sine function. Despite similar $\mathop{\mathrm{PE}}\nolimits$ values across resolutions, $\mathop{\mathrm{LWPE}}\nolimits$ distinguishes coarse approximations (middle) from faithful ones (right), reflecting differences in absolute topological scale.
• Figure 3: Structure and parameters of $\mathop{\mathrm{BNN}}\nolimits_{\sigma}$ in the case of $\sigma=\{a,b\}$, with $a<b\in\mathbb{R}$, $x\in\mathbb{R}$. Each hidden neuron is denoted as $h_{ij}$, where $i$ indicates the hidden layer and $j$ the neuron index within that layer (e.g., $h_{22}$ is the second neuron of the second hidden layer). The numbers written above the neurons correspond to biases, while the labels below indicate the activation functions. The numbers on the edges represent weights. The first hidden layer only aims to compute the barycentric coordinate $t$ within the simplex $\sigma$ , while the subsequent layers determine the contribution of the point within that simplex.
• Figure 4: Comparison of function approximation using a $\mathop{\mathrm{BNN}}\nolimits$ with 8 base points, optimized via two persistent entropy-based loss functions. (a) Initial state with randomly initialized base points. (b) 50th iteration, obtained using the standard persistent entropy-based loss $L_{\mathop{\mathrm{PE}}\nolimits}$. (c) 50th iteration, obtained using the length-weighted persistent entropy-based loss $L_{\mathop{\mathrm{LWPE}}\nolimits}$. (d) MSE learning curves showing convergence differences between $L_{\mathop{\mathrm{PE}}\nolimits}$ and $L_{\mathop{\mathrm{LWPE}}\nolimits}$.
• Figure 5: Similar point-cloud-based function consisting of 250 points as in Fig. \ref{['fig:entropyCalculation']}, but with noise (left). The persistence barcode (middle) and its filtered version (right), showing only the 4 most significant bars, which illustrate the topological features that can be approximated using 8 base points to construct the $\mathop{\mathrm{BNN}}\nolimits$.

Theorems & Definitions (13)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Lemma 1
• proof
• Remark 4
• Definition 2
• Example 1
• Theorem 1