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
Famous quotes containing the words formal and/or definition:
“The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)