Unfortunately, nested comments can't go on further than a depth of 6.

I just wanted to add that other techniques are taking data and using the Veronese embedding and applying Euclidean distance there. The idea here is that you have some additional info on your data, maybe it's only 20 variables and degree 8 or something, and then you can map these points under this embedding and consider the distance here. Alternatively, you can consider the inverse of this problem where you are trying to find some (approximate, possibly) generator of the ideal that defines the points on the variety.