# Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction

arXiv: https://arxiv.org/abs/2512.03899 · HTML: https://mlsanity.com/paper/2512.03899/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## Builds on (most cited first, 11 of 11)

- [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)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [From t-SNE to UMAP with contrastive learning](https://mlsanity.com/paper/2206.01816/) (2022)
- [Insights into Ordinal Embedding Algorithms: A Systematic Evaluation](https://mlsanity.com/paper/1912.01666/) (2019)
- [Uniform Manifold Approximation and Projection (UMAP) and its Variants: Tutorial and Survey](https://mlsanity.com/paper/2109.02508/) (2021)
- [Functorial Manifold Learning](https://mlsanity.com/paper/2011.07435/) (2020)
- [Towards One Model for Classical Dimensionality Reduction: A Probabilistic Perspective on UMAP and t-SNE](https://mlsanity.com/paper/2405.17412/) (2024)
- [Flattening Multiparameter Hierarchical Clustering Functors](https://mlsanity.com/paper/2104.14734/) (2021)

## Abstract (arXiv)

Fuzzy simplicial sets have become an object of interest in dimensionality reduction and manifold learning, most prominently through their role in UMAP. However, their definition through tools from algebraic topology without a clear probabilistic interpretation detaches them from commonly used theoretical frameworks in those areas. In this work we introduce a framework that explains fuzzy simplicial sets as marginals of probability measures on simplicial sets. In particular, this perspective shows that the fuzzy weights of UMAP arise from a generative model that samples Vietoris-Rips filtrations at random scales, yielding cumulative distribution functions of pairwise distances. More generally, the framework connects fuzzy simplicial sets to probabilistic models on the face poset, clarifies the relation between Kullback-Leibler divergence and fuzzy cross-entropy in this setting, and recovers standard t-norms and t-conorms via Boolean operations on the underlying simplicial sets. We then show how new embedding methods may be derived from this framework and illustrate this on an example where we generalize UMAP using vCech filtrations with triplet sampling. In summary, this probabilistic viewpoint provides a unified probabilistic theoretical foundation for fuzzy simplicial sets, clarifies the role of UMAP within this framework, and enables the systematic derivation of new dimensionality reduction methods.