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 c

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.