# Kan Extensions in Data Science and Machine Learning

arXiv: https://arxiv.org/abs/2203.09018 · HTML: https://mlsanity.com/paper/2203.09018/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 17 Mar 2022.

## Builds on (most cited first, 7 of 7)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Algebraic Stability of Zigzag Persistence Modules](https://mlsanity.com/paper/1604.00655/) (2016)
- [Algebraic Data Integration](https://mlsanity.com/paper/1503.03571/) (2015)
- [Edit Distance and Persistence Diagrams Over Lattices](https://mlsanity.com/paper/2010.07337/) (2020)

## Built on by (most cited first, 3 of 3)

- [Using Enriched Category Theory to Construct the Nearest Neighbour Classification Algorithm](https://mlsanity.com/paper/2312.16529/) (2023)
- [Learning Is a Kan Extension](https://mlsanity.com/paper/2502.13810/) (2025)
- [Learning Transfers: Kan Extensions of Topological Invariants of Learned Representations](https://mlsanity.com/paper/2606.07627/) (2026)

## Abstract (arXiv)

A common problem in data science is "use this function defined over this small set to generate predictions over that larger set." Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science. We begin by deriving a simple classification algorithm as a Kan extension and experimenting with this algorithm on real data. Next, we use the Kan extension to derive a procedure for learning clustering algorithms from labels and explore the performance of this procedure on real data. We then investigate how Kan extensions can be used to learn a general mapping from datasets of labeled examples to functions and to approximate a complex function with a simpler one.