Intrinsic Dimension - Generalizations

Generalizations

The type of intrinsic dimension described above assumes that a linear transformation is applied to the coordinates of the N-variable function f to produce the M variables which are necessary to represent every value of f. This means that f is constant along lines, planes, or hyperplanes, depending on N and M.

In a general case, f has intrinsic dimension M is there exist M functions a₁, a₂, ..., a_M and an M-variable function g such that

• f(x) = g(a₁(x),a₂(x),...,a_M(x)) for all x

• M is the smallest number of functions which allows the above transformation

A simple example is transforming a 2-variable function f to polar coordinates:

• f(x₁,x₂) = g((x₁2 + x₂2)1/2), f is i1D and is constant along any circle centered at the origin

• f(x₁,x₂) = g(arctan(x₂ / x₁)), f is i1D and is constant along all rays from the origin

For the general case, a simple description of either the point sets for which f is constant or its Fourier transform is usually not possible.