What's a manifold, James, and how does it contain or describe the
subset of nxn pixels?
It IS the subset of the bigger set.
A manifold is a subset with an extra property -- it behaves, at each point, similarly to a
Euclidean space. If we create a generative mapping from a low-dimensional Euclidean space
to the larger NxN pixel space (which has N^2 dimensions), and every point in the
low-dimensional space corresponds to a "well-formed" image, then that
establishes the property.
People don't really care about the property in the strict sense, more about the fact
that a low-dimensional Euclidean space is useful for generative purposes e.g. random
sampling in it. So we might use the term "manifold" informally but it
communicates the right idea.