FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles

TL;DR

FibeRed introduces a novel approach to dimensionality reduction for topologically complex data by modeling datasets with vector bundles, preserving large-scale topology while reducing local dimensions, demonstrated on dynamical systems and chemistry data.

Contribution

The paper formalizes the use of vector bundles for topologically aware dimensionality reduction and presents an algorithm that integrates local linear reductions with global topology.

Findings

Learned topologically faithful embeddings in lower dimensions.

Outperformed existing metric-based algorithms on complex datasets.

Effectively captured large-scale topology in reduced representations.

Abstract

Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view, and, as an application, we describe an algorithm which takes as input a dataset together with an initial representation of it in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations, obtained through local linear dimensionality reduction, along the initial global representation. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.