One-to-one transformations do not change the
information content of the data but they are important for
feature conditioning prior to PDF estimation.
For 1:1 transformations,
the J-function (2.3) reduces to the absolute value
of the determinant of the Jacobian matrix (2.24),
One-to-one (invertible) transformations
The PDF projection theorem (2.2) may be thought of as a generalization of
the well-known change of variables theorem from basic probability.
is an invertible and differentiable
multidimensional transformation. Then,
determinant of the Jacobian matrix of the transformation