Fiber Bundle Networks: A Geometric Machine Learning Paradigm

TL;DR

FiberNet introduces a geometric machine learning framework using fiber bundles and Riemannian metrics, enabling interpretable classification through geometric optimization and prototype-based decision regions.

Contribution

It presents a novel fiber bundle-based approach that combines differential geometry with machine learning for interpretable classification.

Findings

Learnable Riemannian metrics identify key frequency features.

Prototype optimization minimizes energy for classification.

Geometric interpretability enhances understanding of model decisions.

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.