The differences comes in the fact that higher dimensional tuples may contain data that is independent of other fields. Say, if you have a tuple that's: (name, dob, address,) and you have a projection function that accepts such a 3-tuple and returns a 2-tuple of (name, dob,). For that function, the address dimension has no relationship at all to the other fields, meaning, that there is not unproject function that a person could create such that 3_tuple == unproject(project(3_tuple)).
With manifolds, the higher dimensions can have a relationship with lower dimensional data, and such a relationship can be encoded into a function. What the research appear to have designed is a system that, given a priori knowledge of task and enough n-tuples for learning, can produce and approximation of such an unproject function.
Thus, after learning, they have a system where, unproject(project(n_tuple)) ~= n+1_tuple. Because they were able to inform the learning system about the nature of the relationship between the two dimensions.