[Question] Is it safe to fit-transform once to get multiple embeddings?

Quick question, a bit of the mathsy side.

Suppose I want to test different embeddings of my data, say in 2, 5 and 10 dimensions; is the UMAP algorithm amenable to fit only once using n_components=10 and then take the first 2 or 5 components, or would this be complete nonsense and I should make separate fittings?

[adelejackson]

(Caveat: I’m not Leland McInnes; I’m a mathematician and am comfortable with the maths behind UMAP but am not familiar with the code. Take this with a grain of salt.)

I wouldn’t expect this to work. Unlike in PCA, the coordinate system in the transformed data has no particular meaning – you get the same cost for any isometry of a given representation (in particular, for a rotation). (There’s also no reason the first two components would work any better than the last two.)

EDIT: The following paragraph is wrong. For example, say you have a 10 dimensional dataset. You can “transform” this under UMAP and you should get basically the same dataset back (up to an isometry, I think). Taking two components of this “transformation”, in general we certainly do not get the embedding you would get with n_components=2 – at least, I would hope not, or you could just restrict to two components instead of running UMAP!

[adelejackson]

I don’t think, however, that you get the “same” dataset up to isometry when n_components = n_features; from what I understood from the paper the umap algo works with local distances, and it attempts to make a reconstruction that preserve the simplicial structure, i.e. it should preserve the topology of the dataset, not its metric.

Yep, you’re completely correct; I forgot that we use the Euclidean metric for the low-dimensional representation, not the knn-weighted one.