What good is placement within a matrix if you can't see what the axes are?
The color dimension is configurable, so it makes sense.
But I don't get what the 3 dimensions of spatial coordinates are, and how to change them.
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Alright my bad it's actually explained in the "tour": https://pair-code.github.io/understanding-umap/
It seems to be some kind of multivariate PCA.
No, UMAP is nonlinear. The general idea is that you generate a neighborhood graph of your data points, do a spectral embedding on that to get your initial result, and then do gradient descent to make its neighborhood graph closer to the high-dimensional one.