Partition of Unity Neural Networks for Interpretable Classification with Explicit Class Regions

2026-01-31

AI Summary

This work proposes Partitioned Unit Neural Networks (PUNN), a novel approach that replaces the conventional softmax-based classification paradigm with explicit probability modeling. Traditional neural networks rely on implicit logit inequalities defined through softmax, yielding decision boundaries lacking interpretability. In contrast, PUNN introduces unit decomposition into classification by directly learning a set of non-negative functions that explicitly represent class probabilities, eliminating the need for a softmax layer. The framework accommodates diverse activation functions—including sigmoid, Gaussian, and bump functions—and supports flexible parameterizations such as standard MLPs or geometric priors like spherical shells, ellipsoids, and spherical harmonics, enabling explicit and adaptable modeling of class regions. Experiments demonstrate that PUNN achieves accuracy within 0.3–0.6% of standard MLPs on synthetic data, UCI benchmarks, and MNIST; moreover, when incorporating geometric priors aligned with data structure, it attains comparable accuracy with up to 300× fewer parameters.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Despite their empirical success, neural network classifiers remain difficult to interpret. In softmax-based models, class regions are defined implicitly as solutions to systems of inequalities among logits, making them difficult to extract and visualize. We introduce Partition of Unity Neural Networks (PUNN), an architecture in which class probabilities arise directly from a learned partition of unity, without requiring a softmax layer. PUNN constructs $k$ nonnegative functions $h_1, \ldots, h_k$ satisfying $\sum_i h_i(x) = 1$, where each $h_i(x)$ directly represents $P(\text{class } i \mid x)$. Unlike softmax, where class regions are defined implicitly through coupled inequalities among logits, each PUNN partition function $h_i$ directly defines the probability of class $i$ as a standalone function of $x$. We prove that PUNN is dense in the space of continuous probability maps on compact domains. The gate functions $g_i$ that define the partition can use various activation functions (sigmoid, Gaussian, bump) and parameterizations ranging from flexible MLPs to parameter-efficient shape-informed designs (spherical shells, ellipsoids, spherical harmonics). Experiments on synthetic data, UCI benchmarks, and MNIST show that PUNN with MLP-based gates achieves accuracy within 0.3--0.6\% of standard multilayer perceptrons. When geometric priors match the data structure, shape-informed gates achieve comparable accuracy with up to 300$\times$ fewer parameters. These results demonstrate that interpretable-by-design architectures can be competitive with black-box models while providing transparent class probability assignments.

Problem

• interpretable classification
• class regions
• neural network interpretability
• partition of unity
• softmax limitations

Innovation

• Partition of Unity
• Interpretable Classification
• Explicit Class Regions
• Geometric Priors
• Softmax-Free Architecture