I came across a very interesting discussion about the mathematical significance of UMAP/tSNE interpretation of data. Of course, for me and most research I look at, the application is clustering of biological samples derived from single cell experiments.

The conclusion in a nutshell: UMAP/tSNE GREATLY overemphasize differences between "clusters", so, after "impressing the reader" with the pretty picture, one has to get down and use other statistical techniques to explore and understand the data. Now I finally understand why cells clearly expressing markers of one very well defined cell population can show up in a completely different UMAP cluster. Because the two clusters are really far closer than represented in UMAP, and are really partially overlapping. So, to all "single cell philosophers" out there: Stop elaborating wondrous theories based on UMAP plots, and go do the validation and start understanding your data.

https://simplystatistics.org/posts/2024-12-23-biologists-stop-including-umap-plots-in-your-papers/