Fiber Bundle Learning: A Topological Framework for Classification Using Homology and Discrete Connections

Many machine-learning tasks involve structured data whose geometry, local feature distributions and global organization interact in ways that are not well captured by existing methods based on vectorization, graph metrics or homological signatures. We introduce Fiber Bundle Learning (FBL), a topological framework that represents each data sample as a discrete fiber bundle and extracts a classification signature combining persistent homology, local feature geometry and gluing structure. FBL builds a base space from the coarse geometry of each object, models local feature patches as fibers and estimates transition maps between neighboring fibers to construct a discrete connection. From this representation, FBL computes a set of invariants: persistent homology of the base, fibers and total space; holonomy obtained by transporting fiber states along cycles; curvature-like quantities measuring transition inconsistency; discrete analogues of characteristic classes. These components are assembled into a fixed-length feature vector that can be used with any standard classifier. We show that FBL yields a signature with three desirable theoretical properties: stability under perturbations of geometry and local features; invariance under isometries and global fiber reparameterizations; robustness to sampling noise. Our synthetic experiments show that FBL distinguishes twisted from untwisted bundles with identical homology, a distinction classical topological methods fail to capture. Additional tests quantify the system’s resistance to noise, its invariance to geometric transformations and the contribution of each signature component. Taken together, our results indicate that representing data through fiber-bundle structure may provide an effective tool for classifying complex, multi-level objects.