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

arXiv: https://arxiv.org/abs/2206.06513 · HTML: https://mlsanity.com/paper/2206.06513/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 64 references that are in the map.
- First posted on arXiv 13 Jun 2022.

## Builds on (most cited first, 10 of 10)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Homology-Preserving Dimensionality Reduction via Manifold Landmarking and Tearing](https://mlsanity.com/paper/1806.08460/) (2018)
- [Approximate and discrete Euclidean vector bundles](https://mlsanity.com/paper/2104.07563/) (2021)
- [Sparse Circular Coordinates via Principal mathbbZ-Bundles](https://mlsanity.com/paper/1809.09269/) (2018)
- [Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations](https://mlsanity.com/paper/1612.02861/) (2016)
- [Coordinatizing Data With Lens Spaces and Persistent Cohomology](https://mlsanity.com/paper/1905.00350/) (2019)

## Built on by (most cited first, 7 of 7)

- [Topological Singularity Detection at Multiple Scales](https://mlsanity.com/paper/2210.00069/) (2022)
- [O(k)-Equivariant Dimensionality Reduction on Stiefel Manifolds](https://mlsanity.com/paper/2309.10775/) (2023)
- [Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus](https://mlsanity.com/paper/2312.04600/) (2023)
- [Windowed Fourier Analysis for Signal Processing on Graph Bundles](https://mlsanity.com/paper/2302.05592/) (2023)
- [Fiber Bundle Networks: A Geometric Machine Learning Paradigm](https://mlsanity.com/paper/2512.01151/) (2025)
- [Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases](https://mlsanity.com/paper/2602.22873/) (2026)
- [Understanding Differentiable Embeddings Through Differential and Integral Geometry](https://mlsanity.com/paper/2608.06809/) (2026)

## Abstract (arXiv)

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.