Papers โบ Fuzzy simplicial sets and their application to geometric data analysis
Fuzzy simplicial sets and their application to geometric data analysis
Lukas Silvester Barth, Hannaneh Fahimi, Parvaneh Joharinad, Jรผrgen Jost, Janis Keck, Thomas Jan Mikhail
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
In this article, we expand upon the concepts introduced by David Spivak about the relationship between the category ๐๐ of uber metric spaces and the category ๐ฌ๐ ๐ฎ๐ณ of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP to apply the theory to manifold learning, dimension reduction and data visualization, while refining some of their constructions. We generalize the adjunction between ๐๐ and ๐ฌ๐ ๐ฎ๐ณ, derive an explicit description of colimits in ๐๐, and show that ๐๐ can be embedded into ๐ฌ๐ ๐ฎ๐ณ. Furthermore, we prove analogous results for the category of extended-pseudo metric spaces ๐๐๐๐๐ญ. We also provide rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and ๐ฌ๐ ๐ฎ๐ณ, which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods (source code on github). Additionally, we developed a new embedding method that allows to preserve clusters detected in the original metric space that we construct from the data. The visualization of the optimization process gives the user information both about the inner-cluster distributions in the original metric space and their inter-cluster relations. We compare our new method with UMAP, Isomap and t-SNE on a series of low- and high-dimensional datasets, demonstrate how our method improves upon them and provide explanations for observed differences.
In Syntology Open this paper in Syntology's Atlas, the map of the papers in Syntology's graph and their citations.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change ยท a person checks every report against the paper or source before anything changes; decisions are listed on /corrections