Formal Definition
For an N-variable function f, the set of variables can be represented as an N-dimensional vector x:
- f=f(x) where x=(x1, x2, ..., xN)
If for some M-variable function g and M × N matrix A is it the case that
- for all x; f(x)=g(Ax),
- M is the smallest number for which the above relation between f and g can be found,
then the intrinsic dimension of f is M.
The intrinsic dimension is a characterization of f, it is not an unambiguous characterization of g nor of A. If the above relation is satisfied for some f, g, and A, it must also be satisfied for the same f and g′ and A′ given by
- g′(y)=g(By)
- A′=B-1 A
where B is a non-singular M × M matrix, since
- f(x)=g′(A′x)=g(BA′x)=g(Ax)
Read more about this topic: Intrinsic Dimension
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