Mathematical Notation

In what follows, the symbol ${\bf x}$ is always used for the (high-dimensional) input data, and the symbol ${\bf z}$ is always used for (lower-dimensional) features, the output of $T({\bf x})$. We view both ${\bf x}$ and ${\bf z}$ as random variables (RV) ^2.1. We will use a simplified notation for a RV and the associated probability density function (PDF). We use the same symbol to represent the RV and a sample of that RV. Distributions are identified by their argument, so $p({\bf x})$ and $p({\bf z})$ are understood to be the distributions of RVs ${\bf x}$ and ${\bf z}$, respectively. When there is the possibility of confusion, we use a subscript, for example, $p_z(T({\bf x}))$ is understood to be the evaluation of the density $p({\bf z})$ at sample ${\bf z}=T({\bf x})$. If there are more than one possible distributions of a given RV, we index them by the statistical hypothesis, such as $p({\bf x}\vert H_0)$, $p({\bf x}\vert H_1), \;\ldots$, etc. For a generic fixed feature PDF, we use $g({\bf z})$.