Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction

2025-12-03

AI Summary

Fuzzy simplicial sets are widely used in manifold learning methods such as UMAP, yet their algebraic-topological definition lacks a probabilistic interpretation, causing a disconnect from mainstream generative modeling frameworks. Method: We propose the first unified probabilistic framework: modeling fuzzy simplicial sets as marginal distributions of probability measures over simplicial complexes; revealing that UMAP’s fuzzy weights correspond to samples from a Vietoris–Rips filtration under random scale; establishing equivalence between KL divergence and fuzzy cross-entropy; and deriving the family of t-norms via Boolean operations on the face poset. Furthermore, we generalize UMAP by integrating the Čech filtration with triplet sampling to enhance modeling flexibility. Contribution/Results: This work establishes a rigorous probabilistic foundation for fuzzy simplicial sets, yielding an interpretable, scalable, and theoretically grounded dimensionality reduction method.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Provides a probabilistic interpretation for fuzzy simplicial sets in dimensionality reduction.
• Clarifies the generative model behind UMAP's fuzzy weights using random Vietoris-Rips filtrations.
• Enables derivation of new embedding methods by generalizing UMAP with Čech filtrations and triplet sampling.

Innovation

• Fuzzy simplicial sets explained as probability measure marginals
• Generative model sampling Vietoris-Rips filtrations at random scales
• New embedding methods derived from unified probabilistic framework