# Functorial Manifold Learning

arXiv: https://arxiv.org/abs/2011.07435 · HTML: https://mlsanity.com/paper/2011.07435/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2020.

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

- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations](https://mlsanity.com/paper/2407.17835/) (2024)
- [Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/2512.03899/) (2025)

## Abstract (arXiv)

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.