> Also, when taking its square root you get a distance

Easy conversion into a distance metric is hugely valuable to making the property amenable to KNN-based dimensionality reduction algos (and I'm sure other things I don't understand, as a non-mathematician)

Here's a library that the creator of UMAP provides (UMAP being a workhorse of dimensional reduction algos), for doing approx nearest neighbor search: https://pynndescent.readthedocs.io/en/latest/api.html#pynnde...