# Fiber Bundle Morphisms as a Framework for Modeling Many-to-Many Maps

arXiv: https://arxiv.org/abs/2203.08189 · HTML: https://mlsanity.com/paper/2203.08189/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 15 references that are in the map.
- First posted on arXiv 15 Mar 2022.

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

- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Augmented CycleGAN: Learning Many-to-Many Mappings from Unpaired Data](https://mlsanity.com/paper/1802.10151/) (2018)
- [Evaluating the Disentanglement of Deep Generative Models through Manifold Topology](https://mlsanity.com/paper/2006.03680/) (2020)
- [Latent Normalizing Flows for Many-to-Many Cross-Domain Mappings](https://mlsanity.com/paper/2002.06661/) (2020)
- [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)
- [Bundle Networks: Fiber Bundles, Local Trivializations, and a Generative Approach to Exploring Many-to-one Maps](https://mlsanity.com/paper/2110.06983/) (2021)
- [Cohomology of Cryo-Electron Microscopy](https://mlsanity.com/paper/1604.01319/) (2016)

## Built on by (most cited first, 2 of 2)

- [Haldane Bundles: A Dataset for Learning to Predict the Chern Number of Line Bundles on the Torus](https://mlsanity.com/paper/2312.04600/) (2023)
- [Graph Vector Field: A Unified Framework for Multimodal Health Risk Assessment from Heterogeneous Wearable and Environmental Data Streams](https://mlsanity.com/paper/2603.28115/) (2026)

## Abstract (arXiv)

While it is not generally reflected in the `nice' datasets used for benchmarking machine learning algorithms, the real-world is full of processes that would be best described as many-to-many. That is, a single input can potentially yield many different outputs (whether due to noise, imperfect measurement, or intrinsic stochasticity in the process) and many different inputs can yield the same output (that is, the map is not injective). For example, imagine a sentiment analysis task where, due to linguistic ambiguity, a single statement can have a range of different sentiment interpretations while at the same time many distinct statements can represent the same sentiment. When modeling such a multivalued function f: X rightarrow Y, it is frequently useful to be able to model the distribution on f(x) for specific input x as well as the distribution on fiber f-1(y) for specific output y. Such an analysis helps the user (i) better understand the variance intrinsic to the process they are studying and (ii) understand the range of specific input x that can be used to achieve output y. Following existing work which used a fiber bundle framework to better model many-to-one processes, we describe how morphisms of fiber bundles provide a template for building models which naturally capture the structure of many-to-many processes.