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Comment by Koshkin | Hacker News Reader

> Data doesn't actually live on a manifold.

Often, they do (and then they are called "sheaves").

Many types of data don’t. Disconnected spaces like integer spaces don’t sit on a manifold (they are lattices). Spiky noisy fragmented data don’t sit on a (smooth) manifold.

In fact not all ML models treat data as manifolds. Nearest neighbors, decision trees don’t require the manifold assumption and actually work better without it.

Any reasonable statistical explanation of deep learning requires there to be some sort of low dimensional latent structure in the data. Otherwise, we would not have enough training data to learn good models, given how high the ambient dimensions are for most problems.

Deep learning specifically yes. It needs a manifold assumption. But not data in general which was what I was responding to.

It turns out a lot of disconnected spaces can be approximated by smooth ones that have really sharp boundaries, which more or less seems to be how neural networks will approximate something like discrete tokens

Can be approximated yes. Approximated well? No, but you can get away with it sometimes with saturation functions like softmax. But badly. It’s like trying to solve an integer program as a linear program. You end up with a relaxation that is not integral and not the real answer.

An integer lattice can only be a manifold in a trivial sense (dimension 0). But not for any positive dimensions.

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