Some theoretical questions on UMAP

[victor-mouquin]

Hi!
After reading the arXiv paper I had a few questions about the theoretical background of UMAP.

1. In Definition 9, is the metric defined that way because we are only interested in 1-skeletons? If $d_i(X_j, X_k) = \infty$ when $j,k \neq i$ then the only way to map non-trivially $FinReal(\Delta^n_{<a})$ for $n \geq 3$ is if $a=0$. It seems then that we are loosing all the information about the metric.

2. When working with higher n-skeletons, what would be the way to glue together the fuzzy simplicial sets? Is there a higher dimensional version of the probabilistic $\mathfrak{t}$-conorm?

3. Would it make sense to try to develop UMAP using higher n-skeletons for small datasets?