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 Čech 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.

Dimension reduction seeks to produce a low dimensional representation of high dimensional data that preserves relevant structure (relevance often being application dependent). Dimension reduction is an important problem in data science for both visualization, and as a potential pre-processing step for machine learning. As a fundamental technique for both visualization and preprocessing, dimension reduction is being applied in a broadening range of fields and on ever increasing sizes of datasets. It is thus desirable to have an algorithm that is both scalable to massive data and able to cope with the diversity of data available. Dimension reduction algorithms tend to fall into two categories; those that seek to preserve the distance structure within the data or those that favor the preservation of local distances over global distance.