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> To play devil's advocate... given that manifolds are so general, wouldn't it be more surprising if natural data didn't form some lower dimensional manifold?

No, because not all data can actually be represented on a topology. It's nice if you can model your data as a normed vector space, but it's not inherently possible in general.

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In a trivial sense, it seems that a Euclidean topology would do the trick, no? I'd be curious to hear counter-examples, of course. In my mind, I suppose what's missing from the "manifold hypothesis", as stated above, is that the manifolds should be more useful than raw data, for example, does their structure respect our notion of object categories or are they sufficiently low-dimensional for visualization?

They key is the low dimensional part not that it is a manifold.

The ambient space may be huge, but data is not spread all over but lies on a tiny subset that can be well described by very few parameters. This limited degree of freedom in the data is what makes it easy to learn. A priori there is no reason why data should have such a property.

I would be interested in your thoughts on how you would map human language understanding to a Euclidean topology.

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