This question is related to this one which was not answered.
The curse of dimensionality states that in high dimension every distance between pairs of points tends to be the same. See this answer for more details. UMAP computes distances in the high dimension space to define the neighbourhood of each point. This neighbourhood is then mapped to a low-dimension one. My question is then, how does UMAP avoid considering all points to be equally spaced and therefore how can it produce separated clusters of points?
I saw that people are using PCA before applying UMAP but the reason for that seems to be more due to computational issues then a theoretical one. The computational cost of dealing with many dimensions and many points was a problem for t-sne but UMAP is said to be able to handle such cases and therefore should not need prior dimension reduction.
Thanks for your help
