# Bundle Networks: Fiber Bundles, Local Trivializations, and a Generative Approach to Exploring Many-to-one Maps

arXiv: https://arxiv.org/abs/2110.06983 · HTML: https://mlsanity.com/paper/2110.06983/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 41 references that are in the map.
- First posted on arXiv 13 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Diverse Image-to-Image Translation via Disentangled Representations](https://mlsanity.com/paper/1808.00948/) (2018)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Mode Seeking Generative Adversarial Networks for Diverse Image Synthesis](https://mlsanity.com/paper/1903.05628/) (2019)
- [C-RNN-GAN: Continuous recurrent neural networks with adversarial training](https://mlsanity.com/paper/1611.09904/) (2016)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [Learning Disentangled Joint Continuous and Discrete Representations](https://mlsanity.com/paper/1804.00104/) (2018)
- [Wasserstein Divergence for GANs](https://mlsanity.com/paper/1712.01026/) (2017)
- [InfoGAN-CR and ModelCentrality: Self-supervised Model Training and Selection for Disentangling GANs](https://mlsanity.com/paper/1906.06034/) (2019)
- [The Geometry of Synchronization Problems and Learning Group Actions](https://mlsanity.com/paper/1610.09051/) (2016)
- [Approximate and discrete Euclidean vector bundles](https://mlsanity.com/paper/2104.07563/) (2021)
- [Cohomology of Cryo-Electron Microscopy](https://mlsanity.com/paper/1604.01319/) (2016)

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

- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [Fiber Bundle Morphisms as a Framework for Modeling Many-to-Many Maps](https://mlsanity.com/paper/2203.08189/) (2022)
- [Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus](https://mlsanity.com/paper/2312.04600/) (2023)
- [Discovering Data Manifold Geometry through Geometric Properties](https://mlsanity.com/paper/2602.02611/) (2026)
- [Teacher-Student Guided Inverse Modeling for Steel Final Hardness Estimation](https://mlsanity.com/paper/2510.05402/) (2025)
- [Fiber Bundle Networks: A Geometric Machine Learning Paradigm](https://mlsanity.com/paper/2512.01151/) (2025)
- [RecBundle: A Next-Generation Geometric Paradigm for Explainable Recommender Systems](https://mlsanity.com/paper/2603.16088/) (2026)

## Abstract (arXiv)

Many-to-one maps are ubiquitous in machine learning, from the image recognition model that assigns a multitude of distinct images to the concept of "cat" to the time series forecasting model which assigns a range of distinct time-series to a single scalar regression value. While the primary use of such models is naturally to associate correct output to each input, in many problems it is also useful to be able to explore, understand, and sample from a model's fibers, which are the set of input values x such that f(x) = y, for fixed y in the output space. In this paper we show that popular generative architectures are ill-suited to such tasks. Motivated by this we introduce a novel generative architecture, a Bundle Network, based on the concept of a fiber bundle from (differential) topology. BundleNets exploit the idea of a local trivialization wherein a space can be locally decomposed into a product space that cleanly encodes the many-to-one nature of the map. By enforcing this decomposition in BundleNets and by utilizing state-of-the-art invertible components, investigating a network's fibers becomes natural.