Fiber Bundle Networks: A Geometric Machine Learning Paradigm

Abstract: We propose Fiber Bundle Networks (FiberNet), a novel machine learning framework integrating differential geometry with machine learning. Unlike traditional deep neural networks relying on black-box function fitting, we reformulate classification as interpretable geometric optimization on fiber bundles, where categories form the base space and wavelet-transformed features lie in the fibers above each category. We introduce two innovations: (1) learnable Riemannian metrics identifying important frequency feature components, (2) variational prototype optimization through energy function minimization. Classification is performed via Voronoi tessellation under the learned Riemannian metric, where each prototype defines a decision region and test samples are assigned to the nearest prototype, providing clear geometric interpretability. This work demonstrates that the integration of fiber bundle with machine learning provides interpretability and efficiency, which are difficult to obtain simultaneously in conventional deep learning.

• The paper introduces Fiber Bundle Networks that recast classification as a geometric problem using fiber bundles and Riemannian metric learning.
• It employs variational prototype optimization to minimize an energy functional, yielding clear decision boundaries.
• Experimental results, including audio classification, demonstrate improved interpretability and computational efficiency in resource-constrained settings.

Fiber Bundle Networks: A Geometric Machine Learning Paradigm

Introduction to Fiber Bundle Networks

The paper "Fiber Bundle Networks: A Geometric Machine Learning Paradigm" (2512.01151) introduces Fiber Bundle Networks (FiberNet), a novel approach to integrate differential geometry with machine learning. This framework diverges from traditional deep learning models by emphasizing geometric optimization on fiber bundles rather than merely utilizing black-box function fitting. In this novel paradigm, the authors reformulate classification as a geometric problem where interpretability and efficiency are achieved through the application of geometric tools like Riemannian metrics and variational prototype optimization.

Core Methodological Innovations

Geometric Model on Fiber Bundles

The FiberNet framework models the classification problem using a geometric structure called fiber bundles. Specifically, it considers the category space as the base space and wavelet-transformed feature vectors as residing in the fibers. This approach allows FiberNet to explicitly express the two-layer structure of "categories-features" and employs geometric optimization tools like Riemannian metrics for variance-weighted classification.

Riemannian Metric Learning

A key component of FiberNet is a learnable Riemannian metric, $G = \text{diag}(a_1, \ldots, a_d)$, where each parameter $a_i$ adapts to reflect the discriminative importance of frequency band features. This is especially valuable for tasks like audio classification, where certain frequency bands may carry more semantic significance than others. This adaptive metric ensures computational efficiency and interpretability and is especially advantageous when compared to traditional Euclidean distance metrics.

Figure 1: Wavelet decomposition binary tree structure. Each decomposition layer halves the frequency range, extracting frequency components from high to low.

Variational Prototype Optimization

FiberNet implements variational prototype optimization via energy function minimization, providing a clear geometric interpretation of classification boundaries. Prototypes are optimized by minimizing an energy functional that includes attachment energy, tension, and repulsion terms. These prototypes serve as representatives of each class in the fiber, ensuring that classification decisions can be readily interpreted by observing which prototype is closest under the Riemannian metric.

Practical Implications and Evaluation

Audio Classification Example

To elucidate the effectiveness of FiberNet, the paper presents an example using audio classification. Traditional methods like CNNs often involve vast parameter sizes and lack interpretability. FiberNet provides a stark contrast by modeling the classification problem in a geometrically structured space, where frequency bands are explicitly weighted, and semantic similarities inform the classification process.

The Efficiency and Interpretability Benefit

By incorporating wavelet transformations, FiberNet efficiently captures signal characteristics admired for being both computationally frugal and readily interpretable. This makes it a promising tool for deployment in resource-constrained environments like smart devices, where traditional deep learning models often struggle due to parameter overheads.

Theoretical and Future Implications

The authors provide theoretical guarantees for their model, demonstrating convergence properties of the proposed optimization framework. This robustness highlights FiberNet not only as a computational solution but also as a mathematical innovation in machine learning. Future research may explore extending this geometric setup to other domains like image and text analysis or experimenting with different types of feature transformations and fiber bundle configurations to enhance the adaptability and scalability of the model.

Conclusion

FiberNet emerges as a novel machine learning framework that harmonizes interpretability with computational efficiency by leveraging the rich structure of fiber bundles and differential geometry. By reformulating classification as a geometric problem, it avoids many pitfalls of traditional deep learning, offering a promising path forward for interpretable, efficient machine learning models across a variety of data types. This contribution lays a foundation for future explorations into geometric machine learning paradigms and their application in real-world scenarios where interpretability and resource efficiency are crucial.