Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction

TL;DR

This paper introduces a probabilistic framework for fuzzy simplicial sets used in nonlinear dimensionality reduction, specifically clarifying UMAP's theoretical basis and enabling the development of new embedding methods.

Contribution

It provides the first probabilistic interpretation of fuzzy simplicial sets, connecting them to generative models and enabling systematic derivation of new reduction techniques.

Findings

Fuzzy weights in UMAP originate from sampling Vietoris-Rips filtrations.

The framework links fuzzy simplicial sets to probabilistic models on face posets.

New embedding methods can be derived from the probabilistic perspective.

Abstract

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 \v{C}ech 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.