用几何结构让分类模型可解释，同时提升效率。
Fiber Bundle Networks: A Geometric Machine Learning Paradigm
- 将类别设为基空间，特征放纤维上，通过几何优化做分类
- 学习重要频率成分的黎曼度量，原型通过能量最小化优化
- 适合需要可解释性的场景，如医疗、金融决策
我们提出纤维束网络（Fiber Bundle Networks, FiberNet），一种融合微分几何与机器学习的新框架。不同于传统深度神经网络依赖黑箱函数拟合，我们将分类问题重新表述为在纤维束上的可解释几何优化：类别构成基空间，小波变换后的特征位于每个类别的纤维之上。提出两项创新：(1) 可学习的黎曼度量，用于识别关键频率特征成分；(2) 通过能量函数最小化实现变分原型优化。分类通过学习到的黎曼度量下的 Voronoi 划分完成，每个原型定义一个决策区域，测试样本被分配给最近的原型，提供清晰的几何可解释性。实验表明，该方法在保持高效率的同时实现了传统深度学习难以兼顾的可解释性。
原文摘要 · Abstract (English)
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.
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