Yeah I made the simplification of random vectors myself just to approximate what uncorrelated "features" in an embedding space could be like.

One thing that's relevant for embedding space size Takens theorem: https://en.wikipedia.org/wiki/Takens%27s_theorem?wprov=sfla1

If you have an originally D dimensional system (measured using correlation or information dimension for example), and you time delay embed data from the system, you at most (can be lower) need 2*D+1 embedding dimensions to ensure no false nearest neighbors.

This sets an upper bound if you use time delays. Now, for a *non-*time delayed embedding, I don't know the answer. I asked GPT4 and it said no analytical method for determining embedding dimension M presently exists ahead of time. An experimental method does exist that you can perform before training a model: You need to grow the number of embedding dimensions M and calculate FNN every time M grows. Once FNN drops to near zero, then you've finally found a suitable M.

One neat part about all this is that if you have some complex D-dimensional manifold or distribution with features that "poke out" into different directions in the embedding space (imagine a wheel hub with spokes), then increasing the embedding space size M will also increase the distance between the spokes. If M gets large enough, all the spokes should be nearly equal in distance from each other, but points along a singular spoke are also far from each other in most directions except for just a small subset.

I don't think that making it super large would actually make learning on the data any easier though. Best to stick with close to the minimum embedding dimension M. If you get larger, then measurement noise in your data becomes more represented in the embedded distribution. These dynamics also unfold when you increase M, which means if you're trying to only predict the D-dimensional system, you'll have harder time because now you're predicting a (D+large#) dimensional system and the obviousness of the D-dimensional system distribution gets lost in the larger distribution.