Intrinsic Dimension - Formal Definition

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, ..., x_N)

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)