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 manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)