Why is the manifold hypothesis true?

The manifold hypothesis is the statement that real-world high dimensional data (such as images) lie on low-dimensional manifolds embedded in the high-dimensional space. It has been tested to be true quite extensively. So could someone explain or give some intuition why it is true? Why doesn't real world data such as dogs form manifolds of same dimension as the image space?

Presumably you don't really get a manifold.

For one thing, there's probably a nonzero chance of any particular coordinate being perturbed arbitrarily -- an impurity in a crystal, a grey hair on a head of brown ones, etc. -- so you'd really get something more like a probability distribution concentrated near a lower dimensional platonic ideal.

Second, there are probably all sorts of behaviors going on that aren't allowed in a manifold, like self-intersections, edges, and components of different dimensions.

That said, simple statements like "the left side of someone's face usually looks about like the mirror image of the right" cut the dimensionality of the space nearly in half. Add enough such independent conditions and you whittle things down quite a bit.

StackExchange Computer Science Q#103256, answer score: 9