One-to-one (invertible) transformations

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),

$$J({\bf x};T) = \vert{\bf J}_T({\bf x})\vert.$$

The PDF projection theorem (2.2) may be thought of as a generalization of the well-known change of variables theorem from basic probability. Let ${\bf z}=T({\bf x})$, where $T({\bf x})$ is an invertible and differentiable multidimensional transformation. Then,

$$p_x({\bf x}) = \left\vert {\bf J}({\bf x}) \right\vert \; p_z( T({\bf x}) ),$$ (2.24)

where $\left\vert {\bf J}({\bf x}) \right\vert$ is the determinant of the Jacobian matrix of the transformation

$${\bf J}_{ij} = \frac{\partial z_i}{\partial x_j }.$$