Data may contain some unknown structure and may seem high dimensional. There exist various schemes for extracting dominant structures and efficiently representing them in 2D. A currently very popular scheme is UMAP. We clarify the underlying mathematics and introduce some new geometric ideas and on that basis develop an improved method. 

Related references:

P.Joharinad, J.Jost, Mathematical principles of topological and geometric data analysis,   Monograph, Math of Data, Springer, 2023

L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck, Data visualization with category theory and geometry, Monograph, Math of Data, Springer, 2025

L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck, IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations, Proc. AAAI Conf. Artificial Intelligence 39 (2025); arXiv:2407.17835, with code

L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck, Fuzzy simplicial sets and their application to geometric data analysis Applied Categorical Structures 33 (2025); arXiv:2406.11154

L.Barth, H.Fahimi, P.Joharinad, J.Jost, J.Keck, Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualization,  Adv.Theor.Math.Physics (2026)