Generation of Samples from $G({\bf x};H_0,T,g)$

Indeed, $G({\bf x};H_0,T,g)$ is a generative model. In this book, we will continually discuss the generative process of creating synthetic samples of ${\bf x}$ by drawing samples from $G({\bf x};H_0,T,g)$. Generation of data from $G({\bf x};H_0,T,g)$ is accomplished using the following process ([3], Section 2.1)

1. Draw a sample ${\bf z}^*$ from $g({\bf z})$,
2. Determine the manifold ${\cal M}({\bf z}^*)$, which is the set of all points ${\bf x}$ that map to ${\bf z}^*$ through transformation $T({\bf x})$:

   $${\cal M}({\bf z}^*) = \{ {\bf x}: \; \; T({\bf x})={\bf z}^*, \; \; {\bf x}\in{\cal X} \},$$ (2.4)

   
   
   where ${\cal X}$ is the set of valid input data samples ${\bf x}$. It is common to call ${\cal M}({\bf z}^*)$ a manifold or level set ^2.2.

3. draw a sample ${\bf x}$ from ${\cal M}({\bf z}^*)$ according to a distribution proportional to $p({\bf x}\vert H_0)$.

Note that drawing a sample ${\bf x}$ from ${\cal M}({\bf z}^*)$ according to a distribution proportional to $p({\bf x}\vert H_0)$ can be regarded as a a posteriori distribution of ${\bf x}$ given ${\bf z}$. But, it is not a proper distribution since all its probability mass exists on ${\cal M}({\bf z}^*)$ which has zero volume, and so must have infinite value. If we restrict our analysis just to the set ${\cal M}({\bf z}^*)$, we can write down a representative distribution, called the manifold distribution $\mu({\bf x};H_0,T,{\bf z})$,

$$\mu({\bf x};H_0,T,{\bf z})=\frac{p({\bf x}\vert H_0) \; \delta(T(... ...f z}) }{\int_x p({\bf x}\vert H_0) \;\delta(T({\bf x})-{\bf z}){\rm d}{\bf x}},$$ (2.5)

where $\delta(T({\bf x})-{\bf z})=1$ when ${\bf x}\in {\cal M}({\bf z})$ and is zero otherwise. Clearly

$$\int_{{\bf x}\in {\cal M}({\bf z})} \; \mu({\bf x};H_0,T,{\bf z}) = 1.$$

Intuitively, the manifold distribution is just a distribution on ${\cal M}({\bf z}^*)$ that is proportional to $p({\bf x}\vert H_0)$.